w73 29 v A CRITICAL SURVEY OF WAVE PROPAGATION AND IMPACT IN COMPOSITE MATERIALS by F. C. Moon prepared for NATIONAL AERONAUTICS AND SPACE ADMINISTRATION NASA Lewis Research Center Grant No. NGR 31-001-267 May 1973 PRINCETON UNIVERSITY Department of Aerospace and Mechanical Sciences AMs Report No. 1103 https://ntrs.nasa.gov/search.jsp?R=19730021203 2018-06-28T21:54:46+00:00Z
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w 7 3 29
v
A CRITICAL SURVEY OF
WAVE PROPAGATION AND IMPACT
IN COMPOSITE MATERIALS
by F. C. Moon
prepared for NATIONAL AERONAUTICS AND SPACE ADMINISTRATION
NASA L e w i s Research Center Grant No. NGR 31-001-267
May 1973
PRINCETON UNIVERSITY Department of Aerospace and Mechanical Sciences
A review of the f i e l d of s t r e s s waves i n composite materials is presented covering the period up t o December 1972. and a summary is made of the major experimental resul ts in th i s f i e ld . models f o r analysis of wave propagation i n laminated, f i be r and pa r t i c l e reinforced composites are surveyed. pulses and shock waves i n such materials are reviewed. A review of the behavior of composites under impact loading is presented along with the application of wave propa- gation concepts t o the determination of impact s t resses i n composite plates .
The major properties of waves i n composites are discussed Various theoretical
Theanisotropic, dispersive and dissipative properties of s t r e s s
NASA-C-168 (Rev. 6-71)
ACKNOWLEDGEMENT
This research was supported by NASA/Lewis Research Grant NGR-31-001-267.
This repor t i s t o be published as a chapter i n Trea t i s e of Composite Mater ia ls ,
Broutman, Krock, Edi tors , Vol. 9 , S t ruc tu ra l Analysis and Design, C . C . Chamis,
e d i t o r , Academic Press .
- iii -
TABLE OF CONTENTS
Acknow 1 e dg emen t
Table of Contents
I. Introduct ion
11. Anisotropic Waves i n Composites
A. Wave Speeds
B . Wave Surf aces
C . Flexural Waves i n Orthotropic Plates
D. Surface Waves
E . Edge Waves i n Plates
F. Waves i n Coupled Composite P la t e s
111. Dispersion i n Composites
A. Pulse Propagation and Dispersion
B . Dispersion i n Rods and Plates
C .
D.
E . Continuum Theories f o r Composites
Dispersion i n a Layered Composite
Combined Material and S t ruc tu ra l Dispersion
F - Variat ional Methods f o r Per iodic Composites
I V . Attenuation and Sca t t e r ing
V. Shock Waves i n Composites
V I . Experiments
V I I . Impact Problems i n Comnosites
A. Introduct ion
B . Analytical Models f o r Impact
C . S t ruc tu ra l Presponse t o Impact
References
Captions f o r Figures
ii
iii
1
6
7
10
13
16
18
20
22
23
25
27
33
34
41
44
49
53
61
61
66
72
80
96
- 1 -
I. INTRODUCTION
S t r e s s waves i n composite materials a re of i n t e r e s t t o t h e engineer
both f o r t h e i r cons t ruc t ive -app l i ca t ion and f o r t he po ten t i a l damage t h a t
can occur when s h o r t durat ion stress pulses propagate i n a s t r u c t u r e .
S t r e s s waves have a cons t ruc t ive use as a d iagnos t ic t o o l t o measure
e l a s t i c p rope r t i e s , search f o r flows and t ransmit information. Such appl i -
ca t ion usua l ly involves waves i n the form of pulses o r u l t r a s o n i c s inu-
so ida l pu l se s . Seismologists have long been i n t e r e s t e d i n t h i s appl ica t ion
of stress waves, p a r t i c u l a r l y the study of waves i n layered media (see e .g .
Ewing e t a l . , 1957; Brekhovskikh, 1960). Early s tud ie s of laminated media
were aimed i n fact a t geophysical appl ica t ions (e .g . Anderson, 1961).
S t ruc tu ra l engineers however usua l ly r e l y on composite mater ia l s t o
sus t a in forces o r loads. When these forces are a r e s u l t of shock o r impact
on the s t r u c t u r e , t he forces w i l l be t ransmit ted through t h e s t r u c t u r e i n
t h e form of stress waves.
s t a t i c o r q u a s i - s t a t i c loads (v ibra t ions) can usua l ly be pred ic ted by
s t r u c t u r a l engineers , rou t ine methods f o r pred ic t ing the path
While the pred ic t ion of stress d i s t r i b u t i o n f o r
of stress pulses through a complicated s t r u c t u r e are not r ead i ly
The anisotropy and inherent ava i lab le ,even f o r homogeneous materials.
inhomogeneity i n composite materials f u r t h e r complicates t h i s problem.
The importance of impact stresses i n composite s t r u c t u r a l design can
b e s t be i l l u s t r a t e d by t h e appl ica t ion of these materials t o j e t engine
fan blades (see Goatham, 1970). In addi t ion t o t h e load requirements imposed
by cen t r i fuga l and v ibra tory fo rces , these blades must be designed t o
withstand t h e s t r e s s e s due t o impact with foreign objec ts such as b i r d s ,
- 2 -
ha i l s tones , s tones , and nuts and b o l t s . The r e l a t i v e ve loc i ty of t he
impacting body t o the blade can be i n the order of 450 meters pe r second
(1500 f t / s e c ) .
small impact times (< 50 1-1 sec) and the i n i t i a l transmission of t h e t o t a l
energy i n t o a loca l region of t he blade. The impact not only induces local
c ra te r ing o r s p l i t t i n g but long range damage away from the impact area can
r e s u l t from the r e f l e c t i o n of s t r e s s waves (spal l ing) from boundaries and
focusing e f f e c t s due t o changes i n blade geometry.
problem of foreign object impact involve
embedded high s t rength meshes and leading edge impact pro tec t ion .
The high speed impact of small ob jec ts r e s u l t s i n very
Solutions of t h e
considerable ingenuity,such as
Impact loads involve two fac to r s which are not considered i n s t a t i c
One i s the speed of propagation of t he stress pulse i n
In s t a t i c problems t h e deformation energy can be
s t r e s s analysis .
t he mater ia l .
d i s t r ibu ted throughout t he s t ruc tu re , but i n impact loading the volume of
energy s torage is l imi ted by t h e speed of propagation of t he waves i n
the mater ia l . For shor t time impact loads, a small amount o f energy i n
a small volume can r e s u l t i n s t r e s s e s which can f r ac tu re o r otherwise
damage the mater ia l .
The speeds of propagation of stress waves f o r a number of composites
a re shown i n Table I with comparable da ta f o r conventional s t r u c t u r a l
mater ia l s . These speeds depend on the d i r ec t ion i n which t h e
wave propagates ,and when the e l a s t i c l i m i t i s exceeded ,depend a l so on the
s t r e s s l eve l . These wave speeds a re motions averaged over a local region
of the composite involving many layers , f i b e r s o r p a r t i c l e s whichever i s
the case. Within each cons t i tuent , of course, t h e stresses propagate as
i n the respect ive homogeneous mater ia l s .
- 5 -
The second d i f fe rence between impact loading and s ta t ic loads i n
design i s the rate of change of s t r a i n .
of s t r a i n have been shown t o exh ib i t d i f f e r e n t s t rength proper t ies
Composites under high r a t e s
(Sierakowski e t a l . , 1970). Often t h i s r e s u l t s i n h igher ul t imate
s t r eng th with increasing s t r a i n rate.
While the f ac to r s of f i n i t e wave t i m e and rate dependent proper t ies
are common t o impact problems i n a l l s t r u c t u r a l materials, t he anisotropy
and inhomogeneity inherent i n composites requi res spec ia l a t t en t ion i n the
design of an impact r e s i s t a n t composite s t ruc tu re .
Anisotropic waves i n so l id s a re familiar t o those i n c r y s t a l physics
and seismology, however, these e f f e c t s are not well known i n s t r u c t u r a l
design where conventional i s o t r o p i c mater ia l s such as aluminum and s t e e l
a re o f t en used. Composite mater ia l s have the unique fea ture t h a t the degree
of anisotropy can be var ied i n t h e mater ia l and hence the ana lys t can
change the d i r ec t iona l d i s t r i b u t i o n of s t r e s s waves i n an impact zone and
perhaps avoid ser ious f a i l u r e o r f r ac tu re (perhaps by a judicious choice
of p ly lay-urp angles) .
The e f f e c t s of boundaries o r d i scon t inu i t i e s i n mater ia l p roper t ies
on s t r e s s waves a re well known (see Ewing e t a l . , 1957). When a s t r e s s
wave encounters a boundary,normal t o the wave f ron t , separa t ing mater ia l s
of d i f f e r e n t dens i t i e s and wave speeds, p , v , t he s t r e s s a t t h e surface
is changed t o
= 5 P 2 V2/(P1 v1 + P v 1 0 2 2
- 4 -
where cr would be t h e stress i n material rcl" i f the boundary were
not present . The product pv is ca l l ed the acous t ic impedence and
depends on the type of wave (e.g. shear o r d i l a t a t i o n a l i n i s o t r o p i c
0
s o l i d s ) .
Thus a wave o r ig ina t ing i n a "sof ter" material i . e . p v p2v2 1 1
always s u f f e r s a stress increase a t a boundary, This is indeed t h e case
f o r many composites e spec ia l ly those involving a compliant matrix, such
as epoxy, and a s t i f f f i b e r such as graphi te , g l a s s o r boron.
Another effect of inhomogeneity is dispers ion. Dispersion o f t h e
average composite motion r e s u l t s i n a d i s t o r t i o n of t h e stress pulse
as it propagates.
r ise time, o r per iod o f t he stress pulse decrease.
contaiaing compressional stresses can develop t e n s i l e stresses
The effects of dispers ion increase as the dura t ion ,
Thus a pulse i n i t i a l l y
as the wave propagates and perhaps induce micro-cracking i n t h e composite.
The l i t e r a t u r e on the subjec t of waves i n composites has expanded
enormously i n t h e p a s t few years and new t h e o r e t i c a l and experimental
r e s u l t s a r e s t i l l being repor ted . This review then can only summarize
the work t o the da te o f this wr i t ing . Also severa l good reviews have
appeared a t t h i s wr i t ing i n which t h e var ious t h e o r e t i c a l models f o r
waves i n composites have been discussed, (Peck, 1971, 1972, Achenbach,
1972).
This chapter w i l l be somewhat t u t o r i a l i n na ture r a t h e r than a
c r i t i ca l review of t h e var ious theo r i e s t o da t e . Instead I w i l l t r y
t o summarize the r e s u l t s t o da t e which seem t o be accepted i n t h e f i e l d
and which might be of use t o t h e s t r u c t u r a l dynamics ana lys t .
- 5 -
In the following sec t ions I w i l l discuss
i ) an iso t ropic waves i n composite s t ruc tu res (without dispersion)
i i ) dispers ion e f f e c t on waves
iii) s c a t t e r i n g and absorption o f waves
iv) shock waves i n composites
v) experimental r e s u l t s
v i ) the e f f e c t s o f impact
For a review of s t r e s s waves i n conventional s t r u c t u r a l mater ia l
see Miklowitz (1966).
- 6 -
11. ANISOTROPIC WAVES I N COMPOSITES
In t h i s sec t ion I w i l l review those aspects of e las t ic wave pro-
pagation i n an iso t ropic mater ia l s which a re re levant t o composites.
When the sca l e of t h e changes i n stress l eve l , ( r i s e d is tance , wave-
length, e t c . ) , i s much l a r g e r than the sizes of t he cons t i tuents
of composites ( f i b e r o r p a r t i c l e diameter, f i b e r spacing, p ly spacing,
e t c . ) t he mater ia l may be t r ea t ed as an equivalent homogeneous e l a s t i c
mater ia l as a f irst approximation. In a homogeneous medium, the wave
speeds a r e r e l a t e d t o the e l a s t i c constants and densi ty by r e l a t i o n s
of the type Pv = C , where C is an e l a s t i c constant. This r e l a t ion
*
2
has led t o the use of wave theory t o determine the e f f ec t ive e l a s t i c
moduli of composite mater ia ls when the wavelength becomes l a r g e r than
the s ize of t h e sca l e of inhomogeneity. Thus the de f in i t i on
where X i s the wavelength; ''a" is a s ize associated with t h e composite
elements (e.g. f i b e r spacing ,and v(x/a) is the phase ve loc i ty f o r a
given harmonic wavelength.
(1955) f o r a laminated medium and by Behrens (1967a) (1967b).
This method has been used by White and Angona
In t h e case of p a r t i c u l a t e composites o r dispers ion strengthened
composites t he equivalent model may be considered as i so t rop ic .
f o r f i b e r composites , laminates , and unid i rec t iona l eu tec to ids , t h e
But
* The exception i s t h e case of a composite p l a t e with bending-extensional
coupling.
- 7 -
equivalent s t r e s s - s t r a i n r e l a t i o n w i l l be an iso t ropic i . e .
kR e - t i j - ‘ijk!L
is the stress tensor and e the s t r a i n tensor . where t i j kR
A. Wave Speeds
The simplest wave t o consider is a plane wave with no ex terna l
boundaries present . For such a wave the displacement has the form
The vec tor 2 defines a plane r e l a t i v e t o the mater ia l axes and
v i s the speed of the wave. When Eqs. ( Z ) , (3) a r e put i n t o the
equations of motion f o r the mater ia l , the following eigenvalue problem
r e s u l t s
(‘ijk!L n k n j - P V k j ) % = 0 (4)
the Kronecker d e l t a . (See, e . g . ‘ i j where p i s t h e equivalent densi ty and
Musgrave (1954, 1970) ,and Kraut (1963)). In summary, f o r each wave
d i r ec t ion the re are th ree d i f f e r e n t waves
Ci) When the v (i) a r e d i s t i n c t , the th ree Polar iza t ion vectors
a re orthogonal. For i s o t r o p i c mater ia l s it is well known t h a t only
two speeds a re d i s t i n c t
- 8 -
These a re respec t ive ly t h e longi tudinal and t ransverse (shear) waves.
For an iso t ropic waves however such charac te r iza t ion is not poss ib le
except along symmetry d i r ec t ions .
For many composites, o r tho t ropic symmetry su f f i ces t o descr ibe
the mater ia l and nine e l a s t i c constants are required.
r e l a t i o n f o r t h i s case i s given by
The s t r e s s - s t r a i n
t 11
t 22
t 33
t 23
t 1 3
t 12
c c c 0 0 11 12 1 3
c c 0 0 22 23
c 0 0 33
G O 44
c o 55
C 61
e 11
e 22
e 33
2e 23
2e 1 3
2e 12
For s t r u c t u r a l appl ica t ions composites a re usua l ly used i n the
form o f rods o r p l a t e s . Consider, fo r example, t he in-plane motion
of a p l a t e with t h e x ax i s normal t o the midsurface, i . e . u = 0 . For wavelengths much l a r g e r than t h e p l a t e thickness we neglect t he
2 2
- 9 -
e f f e c t s of dispers ion. For t h i s case the equations of
motion fo r the in-plane motion i n the lowest approximation became
where v i = 1/S P , are t h e e las t ic compliances of t h e
composite. Pot t inger po in ts out t h a t t h e accuracy of t h i s approxi-
mation depends on the values of u t and a/A. For 2% deviat ion from t he exact
dispers ion r e l a t i o n , 2a/A < 0.6 f o r u t = 0.1, and 2a/A < 0.16 f o r V ' = 0.4.
and Sij 1 1
If t h e f i b e r d i r ec t ion of a unid i rec t iona l composite i s var ied
r e l a t i v e t o t h e rod axis , d i f f e r e n t dispers ion r e l a t i o n s are obtained
f o r each angle.
composite.
An example is shown i n Figure 10 f o r a Boron-Aluminm
For a longi tudinal wave i n a p l a t e one can der ive a similar d i s -
pers ion r e l a t i o n f o r
i n e r t i a . For a wave
normal t o the p l a t e ,
anis o t rop i c p 1 a t e s i n co rpo ra t ing the t ransverse
propagating i n t h e x d i rec t ion , and x
t h e phase ve loc i ty is given f o r long wavelengths 1 2
,. where C w a s defined i n E q . ( 7 ) , and d is t h e p l a t e thickness .
11
The Eqs. (24) , (25) neglect t he mater ia l d i spers ion due t o t h e
inhomogeneous na ture of t h e composite, which becomes increasingly
important as the wavelength approaches the scale of t h e s ize of t h e
cons t i tuents . Such e f f e c t s are considered i n the next sec t ion .
- 27 -
C. Dispersion i n a Layered Composite
We now examine the propagation of e las t ic waves i n a s o l i d made up
of a l t e rna t ing layers of d i f f e ren t material s t i f f n e s s e s and dens i t i e s .
This model.has been used by many authors t o examine the e f f e c t s of
dispers ion i n composites (Peck and Gurtman, 1969; Sun, Achenbach, Herrmann,
1968).
authors have given a t t en t ion (Rytov, 1955). The wave concept f o r t h i s
system i s the same f o r connected d i sc re t e p a r t i c l e chains as described
by Br i l lou in (1963).
I t i s a l so an important problem i n seismology t o which numerous
A c e l l is defined as two adjacent l aye r s . A loca l c e l l coordinate
rl
pos i t ion t o any p a r t i c l e i n the composite is given by
a i s the c e l l length, a = d + d . A displacement wave i n the direc-
t i o n normal t o the layer ing has the form
w i l l be used t o d is t inguish one c e l l p a r t i c l e from another. The
x = na + rl where
1 2
We can consider e i t h e r longi tudinal waves f o r which u represents
a displacement normal t o t h e layer ing o r a t ransverse wave where u repre-
s en t s t he displacement p a r a l l e l t o t he layer ing.
0
the longi tudinal speed of sound i n t h e mater ia l .
w i l l represent t h e shear s t r e s s and
the mater ia l .
In the former case l e t
be t he normal s t r e s s on any plane p a r a l l e l t o t h e layer ing and c
For t h e t ransverse case cr
c t h e speed of a shear wave i n
The balance of momentum is given by t h e following equation
- 28 -
along with t h e s t r e s s - s t r a i n r e l a t ion ;
These equations must be s a t i s f i e d i n each of t he two materials and
the so lu t ions i n each l aye r must satisfy boundary conditions of cont inui ty
of s t r e s s vector and displacement. Solutions which s a t i s f y the above
equations and t h e cell boundary conditions a t 11 = 0 are given by
wQ + B s i n a] ikna U (rl) = e [A cos - In C C
1 1
WQ
C 2
B s i n -1 wn p 1 c o l [A COS - + ikna C P C
u (11) = e 2n 2 2 2
- i w t (The f a c t o r e has been dropped f o r convenience). [The term pc is
ca l l ed the acous t ic impedence and i s proportional t o the r a t i o of stress/
ve loc i ty and is analogous t o the same concept in e l e c t r i c a l systems.]
The boundary conditions a t t he nth c e l l - (n + l ) th cel l i n t e r f ace
y i e l d two homogeneous a lgebra ic equations f o r A, B and a l so gives
the dispers ion r e l a t i o n .
wd2 s i n - s i n - wdl w d 2 (1 + P2) cos - - U d l cos ka = cos - C C 1 2
C C 2P 1 2
- 29 -
This r e l a t i o n is pe r iod ic i n k and symmetric about t he k = 0
ax i s . For each k i n the Br i l lou in zone, -IT < k a 2 IT t he re -
are an i n f i n i t e number o f values f o r w and hence an i n f i n i t e
number of branches. There is one acous t ic branch f o r e i t h e r t ransverse
o r longi tudinal waves and may be obtained f o r long wave lengths by
expanding Eq. (26) about k = 0 , w = 0 ;
w = V k 0
where
2 2 (l+p2) d ld2
L = V [-+I + 1-q + 2p a 2 c c 1 2 0
When the acous t ic impedances a re equal
v c C 0 1 2
which is j u s t t h e sum of the times f o r a wave t o t r ave r se each of
t he layers i n t h e cel l .
- 30 -
For t h i s case the waves a re non-dispersive. This i s t r u e because the re
a re no r e f l e c t i o n s a t an i n t e r f a c e of two mater ia l s when t h e acous t ic
impedances a re matched.
If the successive branches a re t r a n s l a t e d t o successive zones,
the dispers ion r e l a t i o n takes t h e character of a continuous homogeneous
medium, as shown i n Figure 11. However there are s top bands pro-
por t iona l t o the mismatch of impedance. This behavior i s a l s o char-
a c t e r i s t i c of quantum e lec t ron waves i n a per iodic p o t e n t i a l of a
conducting s o l i d .
For long wavelengths the d ispers ion r e l a t i o n Eq. (26) can be
expanded about w = 0 , k = 0 ,
r e l a t i o n f o r t he lowest o r "acoustic" wave mode.
t o obtain an approximate dispers ion
v v (1 - a(kd)2) 0 -
where
and
6 = d /a , 6 = d /a. 1 1 2 2
Note t h a t when p = 1, a = 0 .
- 31 -
The long wavelength phase ve loc i ty can be shown t o be r e l a t ed t o t h e
equivalent s ta t ic homogeneous e l a s t i c constant f o r the mater ia l , i . e .
f o r longi tudinal waves
v; = c /p 1 1
p = p v + p v 1 1 2 2
and
where , V = d / (d l+d 1 V = d / (d l+d ) 1 1 2 2 2 2
In the terminology of Herrmann and Achenbach (1968)
t o the e f f e c t i v e modulus,
s t i f f n e s s which i s frequency o r wavelength dependent.
vo i s r e l a t ed
whereas v is r e l a t e d t o an e f f e c t i v e 1
More general ly consider a composite made up of th ree dimensional
repeat ing c e l l s , such t h a t
is a la t t ice vector between corresponding points i n any two c e l l s .
vectors a re ca l l ed a bas i s set f o r t h e mater ia l . One can think of the
mater ia l p roper t ies p , Ci j as per iodic functions i . e . p (x) = P (x + ,&I.
I t is well known t h a t wave-like so lu t ions e x i s t f o r such a medium of
the form
& = Rel + m@2 + ne ( R , m, n , in tegers ) 3
The
- 32 -
Thus the motion var ies by a constant phase f ac to r from cell t o c e l l
and the problem is reduced t o f inding the motion i n a s ing le cel l
i . e . U (r) defined by & = 0 . Further i f one wr i tes 'Lo 'L
i k * r 'LO u ($1 = e 'L 'L ,wo(x)
The function !(g) must be per iodic ,
y0ca + &I = ,wo($)
For f i b e r composites i n two dimensiona, per iodic arrays one would
have only two bas i s vectors i n t h e plane normal t o the f i b e r d i rec t ions
but s imi l a r proper t ies on 8 would obtain.
Other work on the laminated composite includes t h a t of Sve (1971a, 1971b) who
examined thermoelast ic e f f e c t s and waves oblique t o the layer ing.
In addi t ion t o t h e layered o r laminated composite, dispers ion i n
f i b e r o r rod reinforced composites has been s tudied. Approximate
solut ions f o r t h i s problem were given by Puppo e t a l . (1968), Haener
and Puppo (1969), Jones (1970) and Ben-Amoz (1971). Jones shows the
phase ve loc i ty f o r longi tudinal waves t r a v e l i n g down the f i b e r s ,
mode dispersed down f o r t h e lowest mode, as i s indicated by experiments.
( h a y e t a l . (1968), Tauchert and Guzelsu (1972)). He a l so ca lcu la tes
- 33 -
t he cutoff frequency of t h e second mode ,for lon
terms of the matrix and f i b e r p rope r t i e s .
D.
n a l w
Combined Material and S t ruc tu ra l Dispersion
As mentioned above, dispers ion due t o s t r u c t u r a l geometry (e.g.
i n rods o r p l a t e s ) and dispers ion due t o material microgeometry (e.g.
f i b e r s i z e and spacing) have been s tudied separa te ly but i n ac tua l
s t ruc turesboth a re present .
these e f f e c t s a r e examined simultaneously is t h e theory of laminated
A class of problems i n which both o f
p l a t e s and s h e l l s . Multi layer p l a t e s have been s tudied by Jones (1964),
Sun and Whitney (1972), Biot (1972), Dong and Nelson (1972) , Scott
(1972), and Sun (1972a). The study of waves i n c i r cu la r ly laminated
rods o r s h e l l s of two mater ia ls has received a t t en t ion from Lai (1968),
McNiven e t a l . (1963) , Armenakas (1965), (1967) Whit t ier and Jones
(1967) and Chou and Achenbach (1970).
One can perhaps hazard a guess as t o t he comparison of t h e two
e f f e c t s on pulse propagation by an appeal t o the head of t he pulse
approximation discussed above. For longi tudinal waves i n a rod o r
p l a t e t he dispers ive r e l a t i o n f o r a non-dispersive mater ia l has t h e
form, f o r long wavelengths
where a i s a s t r u c t u r a l thickness o r diameter var iab le . The constant
v is r e l a t e d t o t h e square root of an elastic modulus.
m a1 i s i i s p as i n a t e o r f i b e r
If t h e 1
- 34 -
the constant v
ness which i t s e l f depends on t h e wavelength
i s r e l a t ed t o t he square root of an e l a s t i c s t i f f - 1
v '" vo[ l - 1
B2 21
where b is a f i b e r diameter o r lamination thickness. The combined
e f f ec t s of both s t r u c t u r a l and mater ia l dispers ion thus have the form
In most composites, b/a << 1 , s o t h a t it would appear t h a t t he e f f e c t
of mater ia l dispers ion, where the s t r u c t u r a l geometries guide the waves ,
can only become important when 6 >> a.
I
The corresponding problem f o r a composite beam has been examined
i n de t a i by Sun (1972) i n which he examines waves i n a laminated beam
using an e f f ec t ive s t i f f n e s s continuum theory assuming t h a t each
layer obeys the Timoshenko beam assumptions.
f o r a t en layered p l a t e with both an exact analysis and an e f f e c t i v e
modulus Timoshenko beam theory. For a l t e rna t ing layers , of shear
moduli i n t h e r a t i o o f 100, he found t h a t the e f f ec t ive modulus
model, based on Voight averaging of t he constants , agreed with the
exact analysis f o r 2 W h > l where h is the t o t a l beam thickness .
For waves o f sho r t e r wave length the e f f ec t ive modulus model deviated
subs t an t i a l ly from t h e microstructure and exact models.
E . Continuum Theories f o r Composites
Sun compared h i s theory
When t h e dimensions of t h e cons t i tuents of a mixture (e.g. f i b e r
- 35 -
diameter, p ly thickness) are much smaller than t h e s t r u c t u r a l dimen-
s ions , t he engineer is of ten s a t i s f i e d with averages o f t he motions
of t he cons t i tuents . In such cases a continuum model may s u f f i c e
t o describe the motion i n which t h e inhomogeneities a re "smoothed
out".
t o describe conventional s t r u c t u r a l materials which have a hetero-
Examples of such are found i n the use of classical e l a s t i c i t y
geneous gra in s t ruc tu re . A similar model f o r laminated composites,
using an e f f e c t i v e modulus theory has already been discussed above
which does not include mater ia l dispersion of waves (Chamis, 1971).
Attempts t o construct continuum descr ipt ions of dispers ion i n
composites have been var ied , bu t there a r e y i n generalytwo bas i c
approaches. The axiomatic method, is character ized by
the assumption of a s tored energy function with c e r t a i n funct ional
dependence on t h e deformation descr ip tors .
var iab les include descr ip tors f o r t he motion of t he microconsti tuents ,
The kinematic
e.g. f i b e r s o r p a r t i c l e s , i n addi t ion t o the average motion
a t a po in t . Examples of t h i s method are given by Mindlin
(1964) , theory of "microstructure i n e l a s t i c i t y " , Eringen (1966 , 1968)
Eringen and Suhubi (1964), theory of micropolar e l a s t i c i t y and micro-
morphic continua respec t ive ly , and a l so a mixture theory approach
by Green and Naghdi (1965).
broad c lass of mater ia ls and i n t h e l inear ized version of these theor ies ,
These theor ies attempt t o character ize a
contain a g rea t number of material constants which must be determined
by experiment. Ozgur (1971) f o r example has used Eringen's micropolar
theory i n an attempt t o descr ibe or thot ropic f i b e r composites. H i s model
- 36 -
uses 30 material constants as compared t o 9 constants f o r classical
or thot ropic e l a s t i c i t y .
d i spers ion phenomena but does not p red ic t dispers ion f o r longi tudinal
waves.
This theory r e s u l t s i n t h e cor rec t shear
In con t r a s t t o t h e f i rs t method the second approach starts
from an assumption o f a knowledge of t h e proper t ies of each cons t i tuent ,
and by averaging, "smoothing" and energy methods t r i e s t o a r r i v e at
a continuum formulation i n which the material constants are known i n
terms of t he p rope r t i e s of t h e cons t i tuents . An example of t h i s method
has been given by Achenbachand Herrmann (1968a) (1968b) i n which t h e
microelements a re f i b e r s embedded i n an e l a s t i c matrix. The f i b e r s
were assumed t o behave as Timoshenko beams.
continuum is assigned two kinematic va r i ab le s , t h e average displacement
a t a poin t x , and the f i b e r r o t a t i o n vec tor which i s independent
of t he vec tor . The r e s u l t i n g theory thus has s i x d i f f e r e n t i a l
equations o f motion t o be s a t i s f i e d a t each poin t .
able t o p red ic t d i spers ion f o r shear waves.
t he f i b e r s and motion t ransverse t o t h e f i b e r s t he following d ispers ion
r e l a t i o n i s obtained
Each po in t i n t h e equivalent
These authors are
For a wave normal along
k2(1 + 5 (kr )2) I . I ,
p" .2 'Lc 4 4
77
where p * is t h e composite dens i ty , C an e f f e c t i v e shear modulus,
rl Ef , r , t h e f i b e r modulus and 4 4
t h e volume f r a c t i o n o f f i b e r , and
- 37 -
r a t i o n respect
polymer matrix t h e r a t i o
dispers ive effect of t h e reinforce
wavelengths much l a r g e r than t h e f i b e r diameter. This model however
does not p red ic t dispers ion f o r longi tudinal waves. This problem
was solved q u i t e successful ly i n a s e r i e s of papers by Sun e t a l . (1968),
Achenbach e t a l . (1968) f o r t h e case of a laminated composite and by
Achenbach and Sun (1972) f o r a f i b e r re inforced composite.
dispers ion r e l a t ions predicted t h e cor rec t phenomenon at low
frequencies, Figure 1 2 . The exact harmonic wave so lu t ion f o r a l t e rna t ing
The r e su l t i ng
e l a s t i c i s o t r o p i c layers was obtained by Sun e t a l . (1968) and compared
with the continuum theory dispers ion r e l a t i o n . These r e s u l t s (Figure 12)
show good agreement a t low frequencies and f o r mater ia ls whose moduli
do not d i f f e r very much.
A similar method has been employed f o r f i b e r composites by Wu (1971).
In these methods, one e s t ab l i shes a loca l cell at each point
continuum containing a f i b e r and p a r t of t he matrix.
3 i n t he
Embedded i n the
c e l l is a loca l coordinate system 6 . There is assumed a t each poin t
x , a loca l o r microdisplacement f i e l d . In the case of Wu t h i s takes
the form
%
f o r i n g f i d matrix. In the
- 38 -
Achenbach and Sun (1972) they assume d i f f e r e n t forms of displacement
f i e l d s f o r f i b e r and matrix mater ia l i n each c e l l .
Thus f o r r < a , c l = r cos 8 , 5 = r s i n e 2
and f o r r > a
This procedure i s repeated f o r neighboring c e l l s and t h e average d i s -
placements along t h e adjoining c e l l boundaries a re matched. In the
model of Wu (1971) t h i s r e s u l t e d i n cons t r a in t equations on t h e loca l
c e l l s t r a i n s
S imi la r r e l a t i o n s a re obtained i n t h e mode, of Achenbach and Sun. The
concept of a loca l cons t r a in t was first introduced i n these theo r i e s
i n t h e e a r l i e r work of Sun, Achenbach and Hermann (1968) f o r t h e laminated
continuum.
To obta in equations of motion i n these methods the loca l displace-
ment Eq. (28) i s put i n t o cons t i t u t ive equations for t h e f i b e r and
- 39 -
matrix. The r e s u l t i n
over t h e cell c
volume t o y i e l d t
A similar procedure is ca r r i ed out f o r the k i n e t i c energy densi ty a t 4
T(z, 4). equations of motion and boundary conditions
Hamilton's p r inc ip l e is then used t o f ind the d i f f e r e n t i a l
sulj ject t o t he cons t ra in ts between xy 4, (e.g. Eq. (29)) and where W
is the work done on t h e boundary.
Grot (1972) has recent ly completed similar work on the f i b e r composite
continuum and has obtained very good agreement with t h e experiments of
Tauchert and Guzelsu (1971), Achenbach (1972) i n another review i n t h i s
series discusses t h e continuum models of composites.
Other work of a similar na ture includes Ben-hoz (1968), Barker (1970),
Bartholomew (1971), Bolotin (1965), Gurtman e t a l . (1971) Hegemier (1972)
and Koh (1970). Several mixture theor ies have been developed i n which the
contains terms proprotional t o the difference (,u (13- cL u12))
between the average displacements of each const i tuent (e.g. f i b
les of t h i s a r e the works of Bedford and Stern (1971),
work,applied t o
s i t e , estimates t an given i n
- 40 -
terms of the f i b e r and matrix proper t ies and geometry.
models have been given by Lempriere (1969) and Moon and Mow (1970) f o r
spher ica l p a r t i c l e s i n a matrix.
Other mixture
Two comments regarding continuum theor ies of composites a re i n
order before f in i sh ing t h i s sec t ion . First , when the mathematical
s t ruc tu re of these ad hoc continuum models a re examined, one notes a
s imular i ty with t h e axiomatic theor ies discussed e a r l i e r .
of Sun e t a l . compares with Mindlins (1964) mic roe la s t i c i ty theory and
Wu's model compares with Eringen's micromorphic thoery.
Achenbach (1968) have discussed the appl icat ion of Cosserat theory of
continua t o composite mater ia l s . While spec ia l ized , these ad hoc
theor ies however have the advantage of predict ing the e f f e c t i v e
mater ia l constants f o r t he composite i n terms of t he mater ia l constants
of the cons t i tuents . This approach enables the analyst t o quickly
check h i s predicted dispers ion r e s u l t s ,
while i n t h e more general theor ies such confirmation is not b u i l t i n t o
the theory.
Thus the model
Herrmann and
The second remark concerns the usefulness of continuum theor i e s .
While it is remarkable t h a t t h e laminate continuum theory of Sun e t a l .
(1968) checked so very well with exact theory, what is more remarkable
is t h a t i n so fa r as wave propagation is concerned the d i g i t a l computer
was s u f f i c i e n t t o provide t h e exact dispers ion r e l a t i o n .
of ana ly t i c methods not withstanding,continuum models of composites w i l l
c e r t a in ly f ind s t rong competition from computer or ien ted methods (such
as the f i n i t e element method) i n the fu tu re .
The e f f icacy
- 41 -
F. Var ia t iona l Methods f o r Per iodic Composites
When the cons t i tuents o f a composite are arranged i n a per iodic
array,such as a laminated medium o r a f i b e r composite with uniform
spacing, t he displacements and stresses under harmonic waves Can
This problem i s ca l l ed
Thus
be represented by per iodic funct ions.
"Floquet theory" i n t h e subjec t of d i f f e r e n t i a l equations.
t he problem i s reduced t o f inding a so lu t ion i n one c e l l . Such
problems have analogues i n s o l i d s ta te theory of e lec t ron waves
i n per iodic po ten t i a l s . The so lu t ion of the Schroedinger equation f o r these
problems by va r i a t iona l methods has been out l ined by Kohn (1952).
The extenst ion of these methods t o per iodic composites has been made
by Kohn e t a l . (1972) , who applied the theory t o a laminated composite.
Wu (1971) has applied the va r i a t iona l method of t he above authors t o
a wave propagation normal t o a per iodic f i b e r composite mater ia l .
Wheeler and Mura (1972) and TobBn (1971), have looked a t similar
problems.
According t o the Floquet theory, wave l i k e so lu t ions t o the
equations of motion i n a pe r iod ic medium are themselves represented
i n terms of per iodic functions
- i w t i (k-x-ut) = U ($)e
0 = !iy.tf)e
- 42 -
where 8 i s a l a t t i c e vector . If one writes the stresses i n the
form
- i w t = 6. .e t i j ij
the equations of motion become
+ p W2Uk = 0 OkR , R
A statement of a va r i a t iona l theorem is as follows; (Kohn e t a l . ,
The problem of f inding so lu t ions t o the equations of motion i n
terms of t he functions
and s a t i s f y t h e displacement and stress vector cont inui ty conditions across
the c e l l and c e l l const i tuent boundaries, is tantamount t o finding the
s t a t iona ry value o f the funct ional
U (%) which a re per iodic i n the l a t t i c e vectors , 0
' C with respect t o a complete s e t of functions
l a t t i c e vectors , continuous and have continuous f irst der iva t ives i n the cell
(ekL
{U 1 ' L O
which a re per iob , in the
i )* is the s t r a i n tensor and ind ica tes complex conjugate).
A
This theorem allows one t o choose a l i n e a r combination of functions
from {U } t o approximate the wave i n the c e l l . The amplitudes of
each of t he functions a r e chosen so as t o extremize the funct ional 0
I[;] . from which one obtains t h e dispers ion r e l a t i o n between
This procedure leads t o a homogeneous s e t of a lgebra ic equations
w and 5 .
- 43 -
There w i l l be as many branches t o t h e r e l a t i o n
approximating funct ions.
a($.) as the re are
While the method can produce a reasonable approximation t o t h e
d ispers ion r e l a t i o n , t h e stresses i n t h e ce l l may not be as accurate
and lead t o discontinuous stress vec tors a t t h e cons t i tuent boundaries ,
(Kohn e t a l . , 1972) Bevilacqua, Lee (1971). However more general
v a r i a t i o n a l schemes can achieve b e t t e r stress determination as well as
obtaining the d ispers ion r e l a t i o n . (See e .g . Nemat-Nasser (1972)).
Lee (1972) has recent ly reviewed such methods f o r pe r iod ic composites
Krumhansl (1970) has appl ied Floquet theory t o the propagation of t r a n s i e n t
stress pulses i n a layered medium and similar work has appeared by
Krumhansl and Lee (1971).
- 44 -
I V . ATTENUATION AND SCATTERING
Attenuation of a propagating wave represents l o s s o f energy, i n
cont ras t t o d ispers ion i n which the wave energy is conserved but
r ed i s t r ibu ted i n a deformed stress pulse . Loss o f energy during
dynamic motion i n composites can be a t t r i b u t e d t o a t least fou r
phenomena; i ) v i s c o e l a s t i c o r a n e l a s t i c effects o f the cons t i t uen t s ,
i i ) wave s c a t t e r i n g , i i i ) microfracture , i v ) f r i c t i o n between poorly
bonded cons t i t uen t s .
been the concept of constrained l a y e r damping of beams and p l a t e s (see
e.g. Kerwin (1959), Yan (1972). In t h i s appl ica t ion a th ree l a y e r
laminate has a h ighly v i s c o e l a s t i c l aye r constrained by two s t i f f e r
e l a s t i c layers . A continuum theory f o r a v i s c o e l a s t i c laminated
composite has been given by Grot and Achenbach (1970), Biot (1972), as
well as Bedford and Stern (1971) using a continuum mixture theory. The
former work does not t reat waves, whereas Bedford and Stern ca l cu la t e
the a t tenuat ion coe f f i c i en t i n terms of t h e v i s c o e l a s t i c p rope r t i e s f o r
a wave t r a v e l i n g along the l aye r s .
One important use of v i s c o e l a s t i c damping has
A s i n acous t ics , t h e effect of inhomogeneities i n a s o l i d is t o
s c a t t e r energy out o f an inc ident wave. If the re i s some order t o t h e
inhomogeneities e .g . a pe r iod ic a r r ay of f i b e r s o r p a r t i c l e s , t h i s
s c a t t e r e d energy can be r e sca t t e red back i n t o t h e wave ( i . e . dispers ion)
o r r e f l e c t e d back t o t h e wave source. To t h e ex ten t t h a t t he inhomo-
genei ty is random, e las t ic energy w i l l be s c a t t e r e d out of t h e incident
wave thus a t tenuat ing t h e pulse . Thus a mixture of e l a s t i c s o l i d s can
appear i n i ts averaged p rope r t i e s t o be i n e l a s t i c . Krumhansl (1972) has
- 45 -
some general remarks on t
theory of c r y s t a l la t t ices .
randomly heterogeneous e las t ic medium with a plane harmonic wave
incident on it.
67) Knopoff and H
A t low frequencies the sca t t e red energy shows the
familiar Rayleigh dependence on frequency i . e . w2 . constructed a model f o r t he s c a t t e r i n g of waves propagating normal
t o the f i b e r s , when both f i b e r and matrix are elastic, and ind ica tes
the poss ib le exis tence of d i s s ipa t ion i n the composite under dynamic
Mok (1969) has
loadings. Recently Christensen (1972) and WcCoy (1972) have examined
a t tenuat ion due t o s c a t t e r i n g and d isorder i n composites.
Chow and Hermans (1971) have examined the i n t e n s i t y of s ca t t e r ed
waves i n a composite by considering the densi ty and e las t ic constants
t o be random var iab les independent of an ax ia l coordinate. The authors
calcuate the s c a t t e r i n g cross sec t ion (which is a measure of t he energy
of t he sca t t e red fie1d)and f i n d the cross-section proport ional t o
(two dimensional Rayleigh s c a t t e r i n g ) . Theoretical d a t a on the cross-
sec t ion f o r longi tudinal and shear waves propagating i n a g lass f ibe r -
epoxy matrix composite are presented.
w2
Moon and Mow (1970) presented a theo re t i ca l model f o r a t tenuat ion
i n d i l u t e p a r t i c u l a t e composites using the dynamics of a s ing le p a r t i c l e
i n an e las t ic medium. When the inhomogeneities are d i l u t e (volume
f r ac t ion , Vf < . l o ) and random, a f i rs t approximation t o t he calcu-
l a t i o n of s ca t t e r ed energy can o f t en be obtained from the mechanics
of a s ing le s c a t t e r e r , (The d i f f r a c t i o n of e las t ic waves by s ing le
s c a t t e r e r s has b 1971)). When the densi ty
of a r i g i d inclusion p , embedded i n an e l a s t i c matr ix , is g rea t e r 2
- 46 -
t h e equation of t r ans - P > than t h a t of the matrix, i . e .
l a t i o n a l motion o f the sphere U can be found t o be 2
(U-U) = 0 (31) 9 ~ 1 ( z K 3 + 1) 9P 1 [s - %j + ( 2 K 2 + 1)
+ - T
d2 U P -
d t2 ( 2 K 3 + 112 0
where u , i s the average motion o f t h e matrix without t h e inc lus ion
K, is the r a t i o of d i l a t a t i o n a l t o shear speed i n the
mart ix , cL’cs T = a/CL
0
a , radius o f t h e sphere
The form of t h i s equation suggests a mixture theory i n which the
e l a s t i c energy depends on ( U - U ) ~ , and a d i s s ipa t ion funct ion pro-
por t iona l t o ( U - U ) ~ . The dependence on the ve loc i ty U accounts . .
f o r t he r ad ia t ion of e l a s t i c energy when the p a r t i c l e v ib ra t e s i n the
matrix. The dependence on the matr ix ve loc i ty fi accounts f o r
s c a t t e r e d waves i f t he p a r t i c l e were motionless. The r e s u l t i n g
f ’ equations f o r t he p a r t i c u l a t e composite, of volume f r a c t i o n V
were found t o be
a 2~ - y - a2u = P2vf - a 2u
1 a t 2 a 2 2 a t 2 P (1 - Vf) -
- 47 -
where
2p T (2K2 + 2 0 2 0
If the damping were neglected the medium would exhib i t a natura.1
frequency of Q (e.g. p /p = 10, K = 2, a = 1 0 - ~ m , cL = 4 103m/sec.,
R /27r % 2 l o 6 HZ) . pulse (wavelength A >> a , and i n i t i a l i n t e n s i t y 1 , a t Z = 0 ,
0 2 1
For N p a r t i c l e s per un i t volume, a s ine wave 0
0
proport ional t o (au/az)2) , w i l l decay as
-N z I = I e
where
The s c a t t e r i n g cross-sect ion y follows the well known Rayleigh
behavior at low frequencies.
While t h i s model is c l e a r l y l imited i n appl icat ion by the
assumptions made, it serves t o make a simple connection between
a t tenuat ion i n a composite mixture of e l a s t i c so l id s and t h e mechanics
- 48 -
of the ind iv idua l cons t i t uen t s . Further work on a t tenuat ion i n
composites i s needed.
Mow t o d i l u t e f i b e r composites could be made using t h e work of Mow
and Pa0 (1971) on the dynamics of a cy l ind r i ca l inc lus ion i n an
e las t ic matrix.
should be taken i n t o account as was done by Mok,and Chow and
Hermans. In a recent paper Sve (1973) constructs an equivalent
v i s c o e l a s t i c model from t h e s c a t t e r i n g of waves by c a v i t i e s i n a
porous laminated composite.
The extension of t h e model of Moon and
For volume f r ac t ions above lO%,multiple s c a t t e r i n g
The above model f o r s c a t t e r i n g of waves is based on the
i n t e r a c t i o n of harmonic waves of wavelengths long compared with
the s i ze o f t he scat terer .
type have been summarized by Mow and Pao (1971).
encounters a stress wave with a very short rise distance, a wave
f ron t ana lys i s based on ray theory may be more e f f i c i e n t . This
method has been employed by Achenbach e t a l . (1968), (1970) and
by Ting and Lee (1969).
t he stress t o rise from zero t o a given value, i n t h e d is tance of
a f i b e r diameter (< .005 inch o r .1 mm) i s o f t h e order - sec.
Such waves only occur i n shock waves o r i n u l t r a s o n i c pulses of
The analyses of problems of t h i s
When t h e scatterer
One should keep i n mind t h a t t h e time f o r
frequency g r e a t e r than 10 HZ.
- 49 -
V . SHOCK WAVES
The previous discussion has assumed t h a t t he deformation i n
t h e propagating waves was small and t h a t t he material behaved i n
a l i n e a r e l a s t i c manner. Nonlinear e l a s t i c wave analyses i n com-
pos i t e s are few, as e .g . t h a t of Ben-Amoz (1971) who s tudied f i n i t e
amplitude waves i n a f i b e r composite f o r waves along the f i b e r s .
Nor has much work been published t o date on p l a s t i c waves i n composites.
Wlodarczyk (1971) has examined shock waves i n p l a s t i c layered media
with l i n e a r unloading behavior.
e l a s t i c - p l a s t i c solid’has been discussed by Johnson (1972) but was
not applied t o composites.
Calculat ion of plane waves i n an iso t ropic
Shock waves i n composites, however, have received a g rea t
deal o f a t t en t ion . In t h i s class of wave phenomena, t he pressures
i n the s o l i d a re assumed t o be SO high , tha t t he mater ia l can be
t r e a t e d as a hydrodynamic f l u i d .
dev ia to r i c s t r e s s e s a r e assumed t o be small compared with t h e mean
s t r e s s o r pressure.
y i e l d o r e las t ic l i m i t s t r e s s .
This means t h a t t h e shear o r
This occurs f o r pressures much g rea t e r than the
A plane shock wave i s defined as a t h i n p lanar region pro-
pagating r e l a t i v e t o the ma te r i a l , across which the ve loc i ty has a
d iscont inui ty . When t h e medium is homogeneous, cont inui ty and
momentum conditions across the shock sur face y i e l d the following
r e l a t ions between t h e densi ty p , normal p a r t i c l e ve loc i ty v , shock
speed U , and pressure P ,
II p(v-U)II = 0
II pv(v-U)II = - IIPII
- 50 -
When the conditions ahead of t h e wave are such t h a t P = 0 , v = 0 ,
the conditions behind the shock requi re t h a t 0 0
P O v 1
u2 = P / P (1 - 1 1 0
In addi t ion one must prescr ibe a cons t i t u t ive r e l a t i o n f o r t he
pressure and s a t i s f y an energy balance across t h e shock.
Munson and Schuler (1970, 1971) have extended t h i s analysis t o
laminated composites and mechanical mixtures.
neglect the thermodynamics and assume cons t i t u t ive r e l a t i o n s f o r a l l
n cons t i tuents i n t h e composite,
i n a l l t he cons t i tuents t o be equal a t any pos i t ion
In t h e i r model they
Pn = Pn(pn) , and requi re t h e pressures
x, i . e .
Pn(x) = P (x) f o r a l l n 1
Applying t h i s theory t o laminates f o r waves t r a v e l l i n g both along
and normal t o t he l aye r s , Munson and Schuler (1970) conclude t h a t t he
shock speed is independent of t he d i r ec t ion , under c e r t a i n assunytions
on t h e s t r a i n i n each cons t i t uen t . The shock speed obtained has the
form
- 51 -
where a: is t h e i n i t i a l volume f r a c t i o n of t h e nth cons t i tuent
and p: t h e i n i t i a l dens i ty . They a l s o conclude t h a t t h e model
i s not l imi ted t o laminates, and can be used f o r any mechanical
mixture.
The p a r t i c l e ve loc i ty immediately behind t h e shock is assumed
t o be equal i n a l l l aye r s and given by
0 0 anPn v = U ( l - c -
1 "n
Thus when the cons t i t u t ive r e l a t i o n s f o r each cons t i tuent are known,
the shock ve loc i ty can be found as a funct ion of t h e p a r t i c l e speed.
This r e l a t i o n i s ca l l ed a Hugoniot curve.
t h i s model t o a mixture o f AR 0 p a r t i c l e s i n an epoxy matrix and
compared t h e i r calcuat ions with experimental po in ts (Figure 1 3 ) . For
t h i s mixture t h e compressibi l i ty is shown t o behave much l i k e t h e
s o f t e r component.
Munson and Schuler appl ied
2 3
Iden t i ca l r e s u l t s were obtained by Torvik (1970). Tsou and
Chou (1970) used a similar model bu t included t h e thermodynamics i n
the ana lys i s .
r e l a t i o n s f o r a multi-continuum.
Measurements of shock waves and shock Hugoniot curves f o r
Bedford (1971) has reported a theory f o r Hugoniot
quartz-phenolic have been performed by I s b e l l e t a l . (1967), Charest
and J e n r e t t e (1969) , and Munson e t a l . (1971). Studies i n shock waves
i n aluminum -polymethyl methacrylate (PMMA) have been reported by
Barker and Hollenbach (1970), and Schuler ( to appear), and Schuler
and Walsh ( t o appear). Other references t o the study o f shock waves i n
composites include Gary and Kirsch (1971) and Holmes and Tsou (1972).
- 52 -
The independence of the shock speed on the d i r ec t ion of pro-
pagation i n an e l a s t i c a l l y an iso t ropic composite can only hold
a t high pressures .
ind ica ted t h a t such dependence on d i r ec t ion has been observed f o r
Munson and Schuler (pr iva te communication) have
some composite systems below pressures of 6 k i loba r s .
The construct ion of t h e o r e t i c a l models i n the region between
e l a s t i c wave theory and hydrodynamic shock model w i l l present a
grea t challenge t o the ana lys t .
- 53 -
VI. EXPERIMENTS
The generation and measu
materials has , i n general , been base
i n applied science used t o study waves i n s o l i d s .
the use of a i r g m s , explosive charges, exploding f o i l f l y e r p l a t e s ,
shock tubes and p i ezoe lec t r i c u l t r a son ic generators. To measure the
These involve
stress waves , s t r a i n gages, p i ezoe lec t r i c c r y s t a l s , capacitance gages,
op t i ca l interferometer , holographic and pholoe las t ic techniques a re
used. Experimental work i n t h i s area, while not as copious as the
theo re t i ca l e f f o r t s , has provided a steady stream of experimental
da ta with which t o check the mathematical models. For a va r i e ty of
mater ia l s , including f i b e r , laminate and woven f i b e r composite, da ta
has been reported on measured wave speeds, a t tenuat ion and dispers ion
of s t r e s s pulses , shock wave behavior, s t r e s s wave induced f r ac tu re
and impact.
The experiments can be categorized by the type o f stress pulse
used.
time, h a l f s ine l i k e pulses induced by p ro jec t ib l e impact, t o shor t
rise time waves induced by explosive f l y e r p l a t e impact. The Fourier
content of these pulses have a la rge zero frequency component.
sonic o r pulsed s ine wave tests have a narrow spectrum centered about
The monotonic compressional pulse has been used,from long rise
Ultra-
a p a r t i c u l a r freque
idea l ly s u i t e d t o map the dispers ion r e l a t i o n d i rec t ly ,by measuring
la t te r waves a re t
v e l o c i t i e s of t he pulses , whil notonic pulse method
d se shape with
- 54 -
passage through t h e material.
One of t he problems associated with using pulsed s i n e waves is
measuring a wave ve loc i ty .
(see e.g. Br i l l ou in , 1960), t he shape o r envelope of t h e pulsed s i n e
wave t r a v e l s a t t h e group ve loc i ty of t h e spectrum center frequency and
As has been pointed out i n many t e x t s
is not equal t o t h e phase ve loc i ty v = w/k when dispers ion i s
present . The v e l o c i t i e s a re r e l a t e d however P --
Bri l lou in a l so descr ibes two o ther v e l o c i t i e s , t h e wave f r o n t ve loc i ty
and the s igna l ve loc i ty . The l a t t e r is associated with t h e f i rs t
a r r i v a l of s igna l s with t h e spectrum center frequency.
ve loc i ty is sometimes equal t o the group ve loc i ty (Br i l lou in , 1960).
The message however is c l e a r ; carefu l de f in i t i on and i n t e r p r e t a t i o n
of u l t r a son ic wave ve loc i ty measurements a re required i n order t o
construct t h e dispers ion r e l a t i o n w(k).
The s i g n a l
Abbott and Broutman (1966) demonstrated t h e use o f a
monotonic pulse t o measure the equivalent e las t ic constants o f steel/
g lass and "S" glass/epoxy composites. This method is v a l i d as long
as the stress rise length and t o t a l pulse length a r e la rge compared
- 55 -
with t h e s i z e of t h e f i b e r s , f i b e r spacing and t h e t ransverse s t r u c t u r a l
dimensions of t h e specimen (e.g. rod diameter o r plate thickness) .
Potapov (1966) used pulsed ultrasound t o measure the e l a s t i c constants
of f ibe rg la s s p l a t e s . He concluded t h a t or thot ropic e l a s t i c i t y gave
a s u f f i c i e n t l y accurate descr ip t ion of t he e las t ic proper t ies s o de te r -
mined by these tests. Markham (1970) a l so used pulsed ultrasound i n
an u l t r a son ic tank t o measure t h e e l a s t i c constants of a carbon f i b e r
epoxy resin composite.
Tauchert and Moon (1970) used the monotonic pulse method and
compared t h e r e s u l t s with da ta from resonance tests and s t a t i c
moduli.
epoxy and glass/epoxy were within 2% f o r waves along the f i b e r s .
was found t h a t t h e wave a t tenuat ion could be predicted from vibra t ion
resonance t 'ests of t h e mater ia l s . Tauchert (1971a, 1971b) has used
The dynamically and s t a t i c a l l y determined moduli f o r boron/
I t
u l t r a son ic waves t o measure a l l t h e e l a s t i c constants of a va r i e ty
of composites. Tauchert (1972) has a l so measured u l t r a son ic a t tenuat ion
i n composites and observed increases i n damping due t o i n i t i a l t e n s i l e
s t r e s s .
Pot t inger (1970) used a s i m i l a r method i n glass/epoxy and boron/
aluminum and found agreement between s t a t i c a l l y and dynamically
determined moduli t o within 3% f o r waves i n bars a t various angles
t o the f i b e r s . Also N e v i l l , Sierakowski e t a l .
method on steel/epoxy bars with waves along t h e s t e e l f i b e r d i r ec t ion .
The increase i n wave speed with volume f r ac t ion of s t e e l checked very
(1972) used the same
' c lose ly the r u l e of mixtures (Figure14 ) . The at tenuat ion was found
t o decrease with increase i n s t e e l .
r e f l e c t i o n from a free end were found t o propagate a t a s l i g h t l y
Tensi le waves generated on
- 56 -
s tud ied by Ross e t . a l . (1972). Other s t u d i e s i n which e 1 as t i c constants
of composite mat e r i a1 s
Tuong (1970) and Cost and Z i m m e r (1970a). Elast ic constants of f i l l e d elastomers
were determined using u l t r a son ic s by Waterman (1966) and showed t h e e f f e c t s of
temperature and percent f i l l e r on the p rope r t i e s of the two phase ma te r i a l s .
Also White and Van Vlack (1970) have used an acous t ic resonance technique t o
determine the p rope r t i e s of open-pore polymer foams with higher-moduli i n f i l t r a t i n g
mat r ices ,
Using a gas dynamic shock (70 p s i ) t o induce a s h o r t use time pu l se ,
Whi t t ie r and Peck (1969) s tud ied the e f f e c t s of d i spers ion i n graphi te and
boron re inforced carbon phenolic composites with t h e wave i n the f i b e r
d i r ec t ions .
wave f r o n t , overshoot, and o s c i l l a t i o n s i n t h e s t r e s s p l a t eau region, which
checked t h e p red ic t ion of Peck and Gurtman (1969).
described i n a paper by Cummerford and Whi t t ie r (1970).
Drumheller (1971a) performed a similar experiment on a laminated composite
The t ransmi t ted pulse showed a smoothed pulse r i s e i n p l ace of t he
This technique is
Lundergan and
of steel and epoxy. They used a f l y e r p l a t e technique t o generate compressional
re again observed.
- 57 -
In another work Lundergan and Drumheller (1971b) s tud ied t h e
impact of an obl iquely laminated composite of steel and polymethyl
methacrylate (PMMA).
responding t o a quasi longi tudina l and a quasi shear pulse , t h e
experiments showed a th ree s t e p s loping wave.
t h a t f u r t h e r work is needed t o explain the discrepancy.
Although theory p red ic t s a two s t e p wave cor-
The authors conclude
e
With somewhat d i f f e r e n t motives Schuster and Reed (1969) used
a f l y e r p l a t e technique t o generate shock waves i n a boron/aluminum
composite a t pressures up t o 76 kbar and impact durat ion of less than
0 . 2 micro sec. The impact ve loc i ty of t he f l y e r plates were increased u n t i l
damage occurred. Increased f i b e r crushing with impact ve loc i ty was
observed and t h e spa l l i ng ve loc i ty was measured f o r aluminum and two
boron/aluminum composites. The s p a l l ve loc i ty f o r t he plasma sprayed,
d i f fus ion bonded composite showed a three f o l d increase i n ve loc i ty
over the s p a l l ve loc i ty f o r t h e aluminum specimens, while t h e plasma
sprayed, brazed composite showed a s l i g h t decrease i n the s p a l l
ve loc i ty compared with aluminum. This dramatic e f f e c t i s a t t r i b u t e d
t o the two d i f f e r e n t geometrical arrangements of t he f i b e r s produced
during f ab r i ca t ion . A s shown i n Figure 15b, i n the d i f fus ion bonded
specimens the f i b e r s a re not touching and hence are able t o a t tenuate
the shock wave by mult iple s c a t t e r i n g . In t h e brazed specimen (Figure
15a) one can see t h a t the f i b e r s a re contacting i n t h e d i r ec t ion of
t he wave.
of an increase i n a t tenuat ion , r e s u l t i n g i n a s p a l l ve loc i ty no
Thus a boron path is created through the medium with less
g rea t e r than t h a t f o r aluminum.
One may conclude from t h i s experiment t h a t t h e f i b e r geometry
w i l l be an increasingly important f a c t o r i n stress wave f a i l u r e as
the stress r i s e d is tance o r pulse length approaches the f i b e r
- 58 -
dimensions. In the experime
and spacing are about 0.1 mm and
i n length i n the aluminum.
Several important papers have examined t h e d ispers ive n o f
composites d i r e c t l y with t h e use of u l t r a s o n i c waves.
(1968) , demonstrated a decrease i n phase v e l
as predic ted by seGeral t h e o r i e s (Peck and Gurtman, 1969) f o r waves
along the f i b e r s of thorne l and Boron reinforced carbon-phenolic composites.
Asay e t a l .
t y with f r e q
The
phase ve loc i ty with frequency out t o 4 MHZ but a change i n t h e r e i n -
forced specimens of Av/v % 0.20 a t 3 MHZ (Figure 16 ) .
carbon-phenolic without t h e f i b e r s showed no change i n
Tauchert and Guzelsu (1972) performed similar experiments on boron/
epoxy and examined a v a r i e t y of wave normal-fiber o r i e n t a t i o n s , f o r
both longi tudina l and shear waves. In addi t ion t o t h e decrease i n
group ve loc i ty of longi tudina l with frequency both across and along
the f i b e r s , shear waves t r ave l ing along t h e f i b e r s and polar ized
normal t o t h e f i b e r s showed a 25% increase i n group ve loc i ty a t about
1 MHZ (Figure 17) . The wavelength i n epoxy at t h i s frequency i s
about 2.6 mm compared with a f i b e r diameter and spacing d is tance o f
about 0 .1 mm.
(1968) i n an e a r l y t h e o r e t i c a l model and by o the r authors (e.g. Sun
This behavior had been predic ted by Achenbach and Herrmann
e t a l . , 1968) n l a te r works. ear waves p r ga t ing across t h e fibers
showed a s l i g h t decrease i n group ve loc i ty with frequency.
e (1972) performed a similar u l t r a s o n i c experi-
s i n e waves on tungsten
They a l s o claim t o have observed a cu tof f band i n frequency as w e l l as
- 59 -
t he second branch of t h e dispers ion r e l a t i o n (opt ica l branch) (Figure 18) .
In addi t ion , a frequency s h i f t i n t ransmit ted pulse lower than t h e in -
c ident pulse frequency w a s observed by these authors as t h e frequency
approached t h e cutoff region. This i s a t t r i b u t e d t o the f i l t e r i n g of
t he higher frequency components i n the pulse lying i n t h e s top bond of
frequencies, e f f ec t ive ly s h i f t i n g t h e observed frequency o f t h e t r ans -
mit ted pulse . I t should be noted t h a t Sutherland and Lingle (1972) claim
t o have measured t h e phase ve loc i ty i n t h e i r r epor t .
d e f i n i t i o n o f t h e ve loc i ty measurement i s lacking and the present
author suspects t h a t t he da ta represent group v e l o c i t i e s .
However prec ise
Rowlands and Daniel (19 72) have used in te r fe rometr ic holography
t o measure t h e t ransverse displacement i n v ibra t ing laminated aniso-
t r o p i c p l a t e s ,
dimensional waves i n an iso t ropic p l a t e s due t o t r ans i en t impact loads.
Dally, Link and Prabhakaran (1971) were able t o observe two
This method may hold some promise f o r observing two
dimensional waves i n or thot ropic f i b e r reinforced p l a t e s using photo-
e l a s t i c i t y . This development was made possible by t h e development
of or thot ropic-b i re f r ingent mater ia ls which were s u f f i c i e n t l y t r ans -
parent f o r photoe las t ic analysis (see Prabhakaran, 1970). In t h i s
study the authors examined both t h e t r ans i en t loading of a h a l f plane
with edge loading and t h e f u l l plane problem with a hole loaded with
an explosive charge of lead ozide,as shown i n Figure 19.
nature of t h e stress wave propagation is c l e a r from the f igu re (moduli
r a t i o EL/ET % 3 . 0 ) . A cusp-like f r inge seen i n Figure 19 might
represent t h e e f f e c t s of shear wave anisotropy.
The an iso t ropic
By measuring t h e
- 60 -
wave sur face of t h e ou te r f r inge t h e authors were ab le t o recons t ruc t
t he ve loc i ty sur face f o r t h e quas i longi tudina l wave of t h e p l a t e
material. This ve loc i ty was within 10% of t h a t determined from t h e
s t a t i c e f f e c t i v e moduli of t h e p l a t e material.
Another pho toe la s t i c s tudy of stress waves is repopted by
Hunter (1970) , who used an explosive s t r i p along t h e specimen edge.
Using a l t e r n a t i n g l aye r s of d i f f e r e n t material, f r inge pa t t e rns
accompanying a plane wave i n t h e layered d i r ec t ion were observed.
Rose and Chow (1971) used a similar method t o observe the build-up
of a s teady wave f r o n t i n a a l t e r n a t e l y layered composite of d i f f e ren t
pho toe la s t i c materials.
Other experimental work on t h e propagation of waves i n composites
includes Benson e t a l . (1970) , Berkowitz and Gurtman (1970) , Berkowitz
and Cohen (1970), Lord (1972). Cohen and Berkowitz ( i n press ) have
s tudied dynamic f r a c t u r e i n composites due t o stress waves-
Sierakowski e t . a l . (1970a, 1970b) have measured dynamic stress s t r a i n -1 r e l a t i o n s f o r var ious composites and s t r a i n rates up t o l o 3 sec
Up t o a 85%
these rates (Figure 20) .
. increase i n u l t imate f a i l u r e stress was observed a t
- 61 -
VII. IMPACT PROBLEMS I N COMPOSITES
A. Introduct ion
The study of impact of i s o t r o p i c so l id s has a la rge l i t e r a t u r e ,
p a r t of which is documented i n the book by Goldsmith (1960).
study of s imi l a r problems f o r composite s t ruc tu res has received
very l i t t l e a t t en t ion a t t h i s wr i t ing .
include the nonelas t ic and nonlinear aspects of t he problem, s ince
the object of such s tud ie s usual ly concerns t h e pred ic t ion o r avoidance
of f a i l u r e due t o impact.
r a t e s of s t r a i n become important, and the inhomogeneity and the
anisotropy i n composite mater ia ls i n v i t e a wider s e t of f r ac tu re o r
f a i l u r e modes. Impact f a i l u r e modes i n i so t rop ic materials include
indentat ion, spa l l i ng , and penetrat ion of t h e p r o j e c t i l e through t h e
s t ruc tu re .
pu l lou t , s p l i t t i n g , and delamination (see Figure 21).
damage, micro f a i l u r e i n the composite could Droduce a loca l stress
r i s e r , change the na tu ra l frequencies , and decrease the fa t igue l i f e .
The
To be real is t ic , one should
Also t h e mater ia l p roner t ies under high
'In composites one must add t o t h i s l ist f i b e r crushing,f iber
Even with no v i s i b l e
Both empirical and ana ly t i c s tud ie s have been made but r a r e l y
a re theory and experiment in tegra ted . Much of t h i s work has been
motivated by the need f o r b i r d and ha i l s tone impact pro tec t ion of
j e t engine composite fan blades.
conducted a t great cos t , have produced r e s u l t s i n t h e form of leading
edge pro tec t ion schemes and in te r leaving steel wire mesh between the
p lys , (Anon, 1971), while ana ly t i ca l s tud ie s have only begun t o explore
the problem (e.g. Moon, 1972).
Empirical s tud ie s of t h i s problem,
- 62 -
One of t h e f irst areas o f i n t e r e s t i n b i r d imnact problems was t h e
design o f a i rcraf t t ransparent s t r u c t u r e s such as windows and wind sh ie lds
t o resist b i r d s t r i k e s . One such discussion is given by McNaughton (1964).
A summary of tes ts i n t h i s a r t i c l e on vinyl sandwich panels reveals t h a t
the pene t ra t ion ve loc i ty V,
wind screens decreased as the cube roo t of the mass of t he b i r d M;
f o r a given set o f s t r u c t u r a l conditions
of one t o e igh t pound b i r d s on a i rcraf t
!W3 = constant .
Research r e l a t e d t o b i r d damage i n a i rc raf t engines has been reported
by Allock and Col l in (1968). Impact by chicken carcases, wax, wood and
ge la t ine dummies have been inves t iga ted f o r t a r g e t shapes resembling b a s i c
geometries. The authors constructed a momentum t r a n s f e r model f o r t he
average impact force F due t o a spher ica l impactor
MV sin20 A t F =
where M i s b i r d mass, V b i r d ve loc i ty , 8 angle o f de f l ec t ion from
l i n e o f f l i g h t ,
diameter t o ve loc i ty . In terms o f b i r d dens i ty p
A t t he durat ion of impact given by r a t i o of p r o j e c t i l e
Measurements showed t h a t t h e assumed impact time was too long and t h e
t h e o r e t i c a l force too low.
round nose t a r g e t s e las t ica l lymounted showed t h a t t h e t a r g e t de f l ec t ion
was proport ional t o t h e b i r d momentum.
Deflections of t h e f l a t p l a t e kn i f e edge and
- 63 -
Research on h a i l impact damage t o t y p i c a l aircraft s t r u c t u r e s
has been presented by Hayduk (1973).
mental and an ana ly t i ca l model €or denting type h a i l damage i n
aluminum fuselage panels o r dome segments (spherical cap).
Comparison i s made of experi-
The range of s t r u c t u r a l impact problems includes o the r phenomena
besides b i r d and ha i l s tone impacts. These include micrometeorite
damage on spacecraf t , dus t , sand and r a i n erosion, and cav i t a t ion
erosion of s o l i d s which involves dynamic stresses due t o col lapsing
bubbles. A discussion of impact erosion by dust p a r t i c l e s f o r metal
sur faces is given by Smeltzer e t . a l . (1970). The mechanics of a
l i q u i d drop impact with a s o l i d surface has been given by Heyman (1969)
and Peterson (1972). Rain erosion o f composites is reported by Schmitt
(1970). B a l l i s t i c problems of high ve loc i ty penetrat ions of p l a s t i c -
aluminum laminates by steel p r o j e c t i l e s have been analyzed by Kreyenhagen
e t . a l . (1970) using numerical computer modes which i l l u s t r a t e severa l
damage modes.
The t e s t i n g of composite mater ia l s under impact forces encompasses
a v a r i e t y o f load and specimen condi t ions.
impact tests use r e l a t i v e l y small beam-like specimens, ( l e s s than 3
inches long) under a t ransverse poin t force . The durat ion of t he
impact is usual ly long compared with the time of a s t r e s s wave t o
t r ave r se the specimen. For example, using a wave speed of v % 5mm/psec,
a length L = SOmm, and pulse time
number
during such tests.
compressive stress pulses of extremely shor t durat ion, (% 0.2 10-6sec)
i n t h i n specimens (%2 nun th ick) are used producing a nondimensional
Classical Izod and Charpy
T = 10-3sec, t he nondimensional
VT/L = l o 2 is a measure of t he number o f r e f l e c t i o n s occurring
In shock wave impact t e s t i n g of composites, high
number, (using t h e same wave speed as above), V T / L % .5 . Also the NASA
- 64 -
and t h e A i r Force i n t h e United S ta tes a re sponsoring b a l l i s t i c impact
tests on composite p l a t e s , as well as f u l l s i z e j e t engine fan blades,
using ha i l s tones and l i q u i d objects t o simulate b i r d impact.
Almond e t a l . (1969) have reviewed t h e l i t e r a t u r e on c l a s s i ca l
impact t e s t i n g on laminated composites. Embury e t a l . (1967) conducted
Charpy V-notch tests on s o f t so lder laminated s t e e l specimens both
with the impact force normal t o t h e laminated surface (crack a r r e s t e r
configuration) and p a r a l l e l t o t h e laminate surfaces (crack d iv ide r
configurat ion) . In the l a t t e r case t h e d u c t i l e - b r i t t l e t r a n s i t i o n
temperature was reduced and t h e specimens showed higher impact energy
absorption over homogeneous s t e e l specimens.
Also Chamis e t a l . (1971) have performed miniature-Izod impact
t e s t s
r e s i n matrix (specimen s ize 7.9x7.9x37.6 mm). The t e s t s included
on f i b e r composites of glass and graphi te f i b e r s i n an epoxy
specimens with the f i b e r s e i t h e r p a r a l l e l o r t ransverse t o t h e cant i lever
longi tudinal ax is . The t e s t s show f a i l u r e modes of cleavage, cleavage
with f i b e r pu l lout , and cleavage with delamination. In t h e t ransverse
mode t h e cleavage included matrix f r ac tu re , f i b e r debonding and
f i b e r s p l i t t i n g .
by these authors t o be cor re la ted with the intralaminar shear s t rength
The t ransverse impact s t rength was found
of the various composites t e s t e d .
In s i m i l a r work, Novak and DeCrescente (1972) repor t t h e r e s u l t s
of Charpy impact t e s t s f o r un id i rec t iona l graphi te , boron, and g l a s s
f i b e r s i n a r e s i n matrix. They conclude t h a t t h e toughness of t he
r e s i n matrix i s not an important f a c t o r i n impact energy absorption.
"S glass" composites showed a higher impact s t rength than boron/resin
- 65 -
and graphi te / res in composites.
mechanisms such as fi lament pu l lou t , shear delamination, ect .
conclude t h a t t he impact s t rength is cor re la ted w i
They a l so evaluate t h e energy-absorption
They
f i b e r s t r e s s - s t r a i n curve.
In a recent paper Peck (1972) has reviewed t h e l i terature on
s p a l l f r ac tu re i n composites using one dimensional shock waves. In
addi t ion t o the work of Schuster and Reed (1969) on f i b e r composi
discussed above, Warnica and Charest (1967) have used
/?
1-2 vsec compres-
s ion pulses on laminated quartz phenolic t o determine s p a l l s t r e s s
thresholds . Similar work by Cohen and Berkowitz (1972) , and Barbee
e t a l . (1970) a re a l so discussed.
I t i s useful t o compare t h e merits of these d i f f e r e n t tests.
s p a l l tests and dynamic s t r e s s - s t r a i n t e s t s (e.g. Sierakowski, e t a l . ,
1971) the stress waves a re one dimensional. Thus, c l e a r l y defined
s t r e s s states are used t o measure t h e mater ia l s t rength proper t ies .
However, i n foreign objec t damage, t h e conditions of impact f a i l u r e
involve the contact o f b lunt ob jec ts with a sur face , thus producing
a complicated stress s t a t e . Izod and Charpy t e s t s appear t o simulate
ac tua l impact, s ince a knife edge on a pendulum encounters a beam-
l i k e specimen. S i m i l a r l y , b a l l i s t i c t e s t s involve a loca l ly inhomogeneous
stress state i n the region of p r o j e c t i l e contact which is found i n
ac tua l impact problems.
In
However, t h e ad hoc nature of these s t r e s s
states does not allow comparison with o the r t e s t s . Thus Izod and
Charpy r a t i n g s of ten cannot be compared. Also, because of t h e small
s ize of t h e specimens and the long contact time (e.g. . ~ 1 0 - ~ s e c ) many
r e f l ec t ions occur du t h e impact thus obscuring the wave l i k e
nature of impact, which might be present i n a lar specimen o r
i n the actual s t r u c t u r e . In b a l l i s t i c tests, however, contact times
of 5 l o m 5 s e c o r less a re obtained f o r high ve loc i ty p r o j e c t i l e s of t h e
- 66 -
order of one inch diameter.
of a p l a t e would r e s u l t i n the t o t a l energy of impact contained i n
a c i r c l e of radius l e s s than 30 cm.
s t ruc tu re fewer r e f l ec t ions might ob ta in than f o r a small t e s t
specimen. I t i s the opinion o f t h i s w r i t e r t h a t s ca l e e f f e c t s are
For a wave speed of 6 mm/Msec the impact
I f t h e p l a t e is p a r t of a la rge
of importance i n impact t e s t s .
l a t ed t o l a rge r s t ruc tu res t h e nondimensional numbers
Thus i f tes t da ta is t o be extrapo-
where T is t h e contact time, v a wave speed, and L a repre-
sen ta t ive length should be matched i n addi t ion t o o ther va r i ab le s .
B. Analytical Models f o r Impact
The t o t a l problem descr ip t ion involves the loca l deformation
a t the impact s i t e and t h e simultaneous determination of t h e motion
of the s t r u c t u r e during and a f t e r impact.
of the s t r u c t u r e takes place over a time period much l a rge r than t h e
impact contact time, and t h e s i z e of t he impactor i s much smaller
than the s t r u c t u r a l dimensions, t he problem may be s p l i t i n t o two
d i s t i n c t p a r t s .
h a l f space, 11) The response o f t he s t ruc tu re t o a prescr ibed loca l
impact force as determined i n Par t I . The e r r o r s involved i n such
a scheme appear t o be on t h e conservative s ide s ince t h e procedure
w i l l underestimate t h e contact time and overestimate t h e contact force
When the overa l l motion
I ) The loca l mechanics of impact with a deformable
- 67 -
(Goldsmith, 1960).
Discussed i n the next s ec t ion i s the impact o f s o l i d objec ts on
s o l i d sur faces . As already mentioned l i q u i d o r r a i n drop impact
erosion is a l s o an important problem.
v e l o c i t i e s , s o l i d s may be t r e a t e d by a hydrodynamic model and,,liquid
drop model may be use fu l .
1.
For s u f f i c i e n t l y high impact a
Impact of a Half Space-Hertz Theory
The problem of an impulsive l i n e force on an an iso t ropic ha l f
space has been given by Kraut (1963) f o r a t ransverse ly e l a s t i c
i s o t r o p i c material. In p a r t i c u l a r , a l i n e source on the surface
normal t o the symmetry ax i s produces two wave sur faces as shown
i n Figure 22 corresponding t o the wave surfaces ,discussed i n an
e a r l i e r s ec t ion .
another e l a s t i c body has not been given t o da te . The waves generated
during a point impact on an i s o t r o p i c h a l f space have been s tudied by
Pekeris (1955) where it was shown t h a t on the surface la rge s t r e s s e s
propagate a t the Rayleigh sur face wave speed.
an e l a s t i c sphere h i t t i n g an e las t ic h a l f space a re not known.
The extension of t h i s work t o dynamical contact with
But t he dynamics o f
Thus without even considering a n e l a s t i c e f f e c t s , t he ana ly t i c l i t e r a t u r e
on dynamic impact i s l imi ted even f o r i so t rop ic materials. Instead,
what has been used is a q u a s i - s t a t i c t h e o r e t i c a l model ca l l ed t h e Hertz theory
(Goldsmith, 1960). This i s based on the s ta t ic deformation produced by
a poin t force on a sur face . When the force , F , i s between
a sphere o f radius R , and a h a l f space, F
- 68 -
is r e l a t e d t o t h e r e l a t i v e approach of sphere and h a l f space, c1 , by
r 1.140 MVg F = s i n 1.068 V o t t 2 -r
01 m
F = 0 , t 7 - r
i
3 / 2 F = K ~
where
( 3 5 )
(v i s Poisson's r a t i o , and E is Young's modulus).
This r e l a t i o n i s nonl inear s ince the contact a r ea na2 depends
on the force .
Equating t h i s force t o t h e change o f momentum o f a sphere during
impact with i n i t i a l ve loc i ty
pressions f o r t he contact time and force h i s t o r y ,
Vo , t h i s theory gives t h e following ex-
2 . 94am a =
vO
(36)
137)
- 69 -
where c1 i s t h e maximum approach, and M i s the mass of t h e sphere. m Extension of t h e Hertz theory of impact t o an iso t ropic bodies has
been made by Chen (1969) and Willis (1966). The contact region has been
shown by Willis t o be e l l i p t i c f o r an an iso t ropic h a l f space i n cont ras t
t o a c i r c l e f o r t he i s o t r o p i c case.
r e l a t i o n similar t o Eq. (35), where K depends i n a complicated way on t h e
e l a s t i c constants .
done numerically and no examples have been given t o date f o r t yp ica l
composite anisotropy.
However he obtains a force def lec t ion
The determination of the e l l i p s e parameters must be
A simple model f o r es t imat ing the contact time f o r i s o t r o p i c spheres
on composites has been suggested by the author Moon (1972d), which assumes
a c i r c u l a r contact a r ea .
on unid i rec t iona l f i b e r composite p l a t e s , with the f i b e r s p a r a l l e l t o
the sur face , show t h e contact area t o be e l l i p i c a l with t h e la rge axis
normal t o the f i b e r s , bu t only s l i g h t l y deviat ing from a c i r c l e .
Experiments on the contact of a s t e e l sphere
Thus i n Eq . (35 t he ha l f space constants (1 - v2)/E a re replaced by a
t ransverse e l a s t i c constant f o r t h e composite. &en* has suggested
using the compliance S where the "3" axis is normal t o the
sur face . The author has used 1 / C t o replace (1 - v2)/E i n
Figure 2 3 t o es t imate the contact time f o r ha i l s tones and g ran i t e
spheres on 55% graphi te f i b e r i n epoxy. For impact speeds i n the
range 100-500 m/sec t h e contact times range from 15-85 usec.
33
3 3
In summary these formulae revea l the following dependence of contact
time and peak pressure on impact ve loc i ty
* Pr iva te communication
- 70 -
Such r e s u l t s , however, should only be used as guidel ines , s ince t h e
theory uses assumptions which break down a t high v e l o c i t i e s . Goldsmith
(1960) has made the following summary of t he l i e r t z theory of contact .
1) A t high v e l o c i t i e s t h e H e r t z contact time is a lower bound on the
contact time.
2) When a sphere s t r i k e s a beam, t h e motion of t h e beam decreases the
force , bu t t h e contact time remains about t h e same.
In another reference Goldsmith and Lyman (1960) have shown t h e Hertz
theory t o be remarkably v a l i d in so fa r as contact time and peak force
f o r the impact of hard s t e e l spheres
s t e e l sur face f o r v e l o c i t i e s up t o 300 f t / sec ( ~ 9 1 . 5 m/sec) . The da ta
i n Figure 2 3 for graphi te epoxy can only be used as a rough guide f o r
contact times, u n t i l experimental d a t a becomes ava i lab le .
2 . Non Hertzian Impact
(1/2 inch diameter) onto a hard
3 / 2 The H e r t z theory of impact rests on the contact law F = ~a
For boron/aluminum and graphi te fiber/epoxy composite p l a t e s t h i s force
law was t e s t e d under 1/4 inch and 3/8 inch s t e e l b a l l s i n a s t a t i c
t e s t i n g machine. The preliminary r e s u l t s i n Figures 24925 show clearly
t h a t a more general l a w i s required and t h a t f o r moderate forces ( l e s s
than 100 l b f ) t h e deformation is i n e l a s t i c , requi r ing a d i f f e r e n t law
f o r approach and rebound.
A more general contact law was given by Meyer (see Goldsmith, 1960)
- 71 -
If such a l a w holds f o r both approach and rebound, formulas similar
t o the H e r t z theory can be obtained, (see Goldsmith, 1960, p . 91) .
Clear ly the s ta te o f knowledge about t he impact of composite o r
inhomogeneous bodies i s unsa t i s f ac to ry .
good q u a s i - s t a t i c theory which can account f o r a n e l a s t i c effects , a
t r u l y dynamical impact model f o r composites i s needed.
In addi t ion t o the lack of a
For i s o t r o p i c mater ia l s computer codes employing f i n i t e d i f -
ference methods have been developed f o r dynamical impact and penet ra t ion
p r o j e c t i l e s and deformable bodies , (e. g. Wilkins , 1969, Kreyenhagen e t . a l . 1970) . These models apply a n e l a s t i c cons t i t u t ive equations and can p red ic t
permanent deformation.
no doubt be ava i l ab le i n the near fu tu re as w e l l as codes based on f i n i t e
element methods. H wever t h e r e i s a need f o r ana ly t i ca l so lu t ion f o r
impact phenomena; f i r s t f o r t h e i r s impl i c i ty and a c c e s s i b i l i t y t o the
designer , and second t o check the computer codes which w i l l c e r t a in ly
appear i n the near f u t u r e .
The extension of these codes t o composites w i l l
In developing a n a l y t i c a l models f o r impact, t h e use of an equi-
va len t an i so t rop ic mater ia l i s quest ionable i f one des i res t o explain
stresses i n t h e contact region. When a composite material i s indented
by another body of convex sur face the a rea of contact goes t o zero as
the contact pressure decreases. Thus f o r small forces t h i s a rea i s
necessa r i ly o f t h e order of t h e dimensions of t h e f i b e r s o r lamina.
One would expect a force-def lec t ion l a w t o exh ib i t pe r iod ic changes
i n s lope as t h e contact area engages each successive f i b e r (Figure 25) .
- 72 -
This i s t h e t e n t a t i v e explanation f o r t he wav ss i n t h e experi-
mental r e l a t i o n shown i n Figure 25 f o r boron f i b e r s
matrix. The pe r iod ic p l a t eau appear t o occur a t de f l ec t ions c
responding t o contact r a d i i d i f f e r i n g by t h e f i b e r spacing (%.004
inches) .
C. S t ruc tu ra l Response t o Impact
1. The Coupled Problem
Further experimental work on t h i s problem i s needed.
When the impact force and durat ion depend on t h e s t r u c t u r a l
motion the above procedure cannot be used. The coupled response
of an i s o t r o p i c p l a t e and a spher ica l impactor was t e a t e d by Eringen
(1953) and o the r s , (see Goldsmith, 1960) . Conceptually t h e extension
t o composite s t r u c t u r e s is similar. Let two coordinate systems be
embedded i n the two bodies (see Figure 26) and le t the axes x , x ' 3 3
be d i r ec t ed i n t o the sur faces of s t r u c t u r e and impactor respec t ive ly .
Relat ive t o these coordinates w , w ' represent sur face de f l ec t ions , 3 3
W , t he de f l ec t ion o f t h e p l a t e o r s h e l l neu t r a l surface, and W f 3 3
t h e displacement of t h e impactor center of mass.
shapes of both s t r u c t u r e and impactor are given by
If t h e sur face
x = S(x ,x ), 3 1 2
x '= S ' (x ' ,x ' ) 3 1 2
then t h e boundary condi t ion t o be s a t i s f i e d over
the contact region i s
on x3, x; = 0
The def lec t ions w w ' are determined from a th ree dimensional 3' 3
ana lys i s , such as a Hertz ana lys i s , (as e .g . Willis, 1966). The
- 73 -
displacement W
theory, while t h e impactor displacement W' is governed by Newtons
law f o r t he body under i n i t i a l condi t ions
i s governed by a two dimensional p l a t e o r s h e l l 3
3
5 W' = 0, - - - V , a t t = O 0 d t 3
The so lu t ion o f such a problem f o r a composite s t r u c t u r e i s not
known t o t h e author , though the problem seems f a i r l y s t ra ightforward.
2 . Transient Load Problems
There has been, however, a number of s tud ie s made of t he response
of a composite body t o sho r t durat ion o r impact-l ike forces . Already
mentioned i s t h e work of Peck and Gurtman (1969) on the response of a
laminated h a l f space t o a compressive stress on t h e sur face i n t h e
d i r ec t ion of t h e layer ing. Sve (1972) has a l so t r e a t e d t h e laminated
h a l f space under impulsive hea t ing of t he sur face , (e.g. from a l a s e r ) ,
with thermoelast ic coupling.
theory of Sun e t a l . (1968). In another work Sve and Whit t ier (1970)
have appl ied t h i s theory t o the pressure loading of an obliquely laminated
h a l f plane t o determine t h e e f f e c t s o f lamination angle and dispers ion
on t h e stresses.
This work uses the approximate continuum
Voelker and Achenbach (1969) t r e a t e d an i n f i n i t e laminated body
under a s t e p body force i n a plane normal t o t h e layer ing using an exact
modal ana lys i s . The i n t e r f a c e shear stress wave shows a slow rise t o
a s t a t i c value, while t he normal i n t e r f a c e stress is found t o be o s c i l l a t o r y .
Also Sameh (1971) has used a d i s c r e t e element model t o ca l cu la t e t h e e las t ic -
p l a s t i c response of a layered h a l f space.
- 74 -
The one dimensional impact loading o f a laminated p la te has been
discussed by Hutchinson (1969) where t h e pressure is normal t o t h e
layer ing .
transmission coe f f i c i en t s f o r a stress pulse when it encounters a
d iscont inui ty . For example, t h e t ransmi t ted stress across a plane
boundary separa t ing two d i f f e r e n t materials with normal stress
inc ident on t h e sur face i s given by
This problem can be solved exac t ly using t h e r e f l e c t i o n and
o = TO T = 2 Z Z / ( Z + z ) 0 ’ 1 2 1 2
where
(Note, t h a t T i s independent of t h e d i r ec t ion of t he inc ident stress).
Thus a pressure d iscont inui ty of i n t e n s i t y p0 propagating normal t o
a laminated medium of a l t e r n a t i n g acous t ic impedances suffers an a t t en -
uat ion at t h e head of t h e pulse of
Z1, Z 2 are t h e acous t ic impedances o f t h e two materials.
a f t e r encountering n p a i r s of l aye r s . Analysis using the r e f l ec t ed
and t ransmi t ted waves i n each l aye r revea ls t he stress h i s t o r y behind
the wave f r o n t .
3 . Transient Edge Loading o f a Plate
As noted ea r l i e r , when the pulse durat ion i s long enough, d i spers ive
effects can b e neglected as a first approximation and an equivalent aniso-
t r o p i c model can be used (Eqs. ( 7 ) , (12) ) . One of t h e effects of anisotrony
- 75 -
i s revealed i n t h e one dimensional edge impact of an o r tho t rop ic
p l a t e with t h e impact force i n the plane of t h e p l a t e and t h e edge
oblique t o a symmetry ax i s
s t r u c t u r a l and material d i spers ion , w e can use Eqs. ( 7 ) , with
Neglecting
t h e boundary condi t ions on t h e edge
For an i s o t r o p i c material a compressional wave would be generated.
However, f o r an edge oblique t o the symmetry a x i s , two waves are
propagated i n t o t h e p l a t e with wave speeds corresponding t o those
on the ve loc i ty sur face with wave normal (cos$, s in$ ) . Also d i s -
placements normal and p a r a l l e l t o t h e edge w i l l be exc i t ed . The
displacements w i l l t ake the form (xn normal t o edge, x along
the edge) n un = u [cos+ - a s i n + ] f ( t - -
S
X
1 1 V 1 n
+ u [cos$ - a s in$ ] f ( t - y 1 2 2 n
X
L n S X
U = U [sin$ + 01 COS$] f ( t - - ) 1 1 V
1 n X
+ u [s in$ + a cos$] f ( t - 7 1 2
2 2
The vectors (1, a l ) and (1, a ) are the eigenvectors corresponding
t o v , v respec t ive ly and depend on t h e angle 4 . 2
1 2
The quasi shear wave is generated through t h e coupling o f t h e
normal stress t.” with t h e shear s t r a i n , e , i n t h e cons t i t u t ive ns
- 76 -
n S n equations wr i t t en i n t h e x - x coordinate system, i . e . on x = 0
The constants
the angle I$ , (see Ashton e t a l . 1969). Determination o f t h e constants
U , U r e s u l t from s u b s t i t u t i o n of Eqs. (41) i n t o these boundary condi-
t i ons and is l e f t f o r t h e reader .
A l l , A I 6 , e t c . a r e r e l a t e d t o the e las t ic constants and
1 2
4 . Impact Generated Flexural Waves
Flexural waves generated by impact forces t ransverse t o i s o t r o p i c
p l a t e s has been reviewed by Mikowitz (1960). The one dimensional l i n e
impact of an iso t ropic p l a t e using both t h e Mindlin Eqs. (12) and t h e
classical theory, Eq. (15), has r ecen t ly been t r e a t e d by Moon (1972d)
In t h i s work t h e l i n e force is t ransverse t o the p la te sur face and
oblique t o t h e composite symmetry axes. In the context of t h e Mindlin
theory extensional waves are generated by a t ransverse force as well as
a f l exura l wave. The importance o f shear deformation and ro t a ry
i n e r t i a , as r e f l e c t e d i n Mindlins theory, i s shown t o become important
when the width of t h e contact force d i s t r i b u t i o n is comparable t o t h e
p l a t e thickness .
The ca l cu la t ion o f t h e two dimensional stress wave response t o
cen t r a l impact forces has r ecen t ly been s tudied by Chow (1971) and ).
Moon (1972b,c). Using a Timoshenko theory f o r laminated o r tho t rop ic p l a t e s
- 77 -
Chow (1971) t r e a t s t h e t r ans i en t response of a rectangular p l a t e t o normal
impact.
The author (Moon 1972b,c) uses a Mindlin p l a t e model t o examine the stress
contours a f t e r impact i n an i n f i n i t e p l a t e .
and f l exura l waves a re shown t o be generated under t ransverse impact.
Again,both extensional
Solutions t o t h e equations which govern the cen t r a l impact of
an iso t ropic p l a t e s were found f o r impact-like pressures using an
analytical/computational method.
used was the following
The impact pressure d i s t r ibu t ion
f o r r < a , ( r 2 = x2 + x2) and t < T 1 3 0
The th ree s t r e s s measures chosen were the average membrane stress
(t + t ) / 2 , t h e average f lexura l stress ( t + t ) / 2 a t the surface 11 33 11 33
of t he p l a t e , and t h e maximum inter laminar shear s t r e s s given by
( t 2 + t 2 21 2 3
The s t r e s s e s were calculated i n a qua r t e r plane of t h e p l a t e f o r a
s p e c i f i c time a f t e r t he i n i t i a t i o n of inipact and were normalized with
respect t o t h e maximum impact pressure as calculated i n the above sec t ion .
The da ta i s presented f o r various times and lay-up angles i n the form
of stress contour p l o t s (Figures 27, 28). Superimposed on these curves
- 78 -
a r e the theo re t i ca l wave f ron t f o r t he p a r t i c u l a r wave i n question
and the radius of t he c i r c l e which bounds the impact pressure.
The s i g n i f i c a n t stress l eve l s a l l l i e within t h e sur face bounded
by t h e theo re t i ca l wave sur face . In Figure 27, the average o r membrane
mean stress contours 1 /2 ( t + t ) f o r graphi te fiber/epoxy matrix
laminate p l a t e s are shown f o r lay-up angles of 1 1 33
O o , k45'.
The f l exura l o r bending motion has th ree waves associated with
i t . The l a r g e s t s t r e s s e s however were found i n t h e lowest f l exura l
wave which t r a v e l s a t an i s o t r o p i c speed given by
(K = lT2/12 ,
mean f l exura l
Figure 28 f o r
lay-up angles)
is Mindlin's cor rec t ion f a c t o r ) . S t r e s s contours f o r t he
stress 1 / 2 ( t - - + t - - ) i n t h i s wave a re shown i n 1 1 3 3
graphi te fiber/epoxy matrix
under the t ransverse impact
the wave f r o n t is c i r c u l a r s ince v is
S t resses i n the second and t h i r d f l exura l 3
0 0 laminate p l a t e s (+15 , 245
pressure E q . (42) . Note t h a t
i s o t r o p i c f o r laminate p l a t e s .
waves were found t o be small.
A t h ree dimensional computer p l o t i s shown i n Figure 29 f o r t he f l exura l
s t r e s s f o r t he k45 lay-up angle composite p l a t e . 0
The maximum stress l eve l s were found t o occur immediately after
the end of impact and appeared t o propagate along the f i b e r d i r ec t ions ,
given by the lay-up angles .
These r e s u l t s show t h e e f f e c t of t he change of f i b e r lay-up angles
on the stress d i s t r ibu t ions . For the f l exura l stresses, t h e optimum
- 79 -
0 lay-up angle t o be 41S0, showing a 34% lower stress l eve l than t h e 445
case. However, regarding t h e inter laminar shear s t r e s s e s , f o r t he same
impact conditions, there seems t o be l i t t l e difference i n t h e maximum
s t r e s s leve l with lay-up angle desp i te s ign i f i can t changes i n s t r e s s
d i s t r i b u t i o n i n space with Pay-up angle.
Another r e s u l t of these calculat ions i s t h a t t he induced s t r e s s e s
depend on t h e impact c i r c l e radius t o p l a t e thickness r a t i o .
O f course, t o evaluate t h e p o s s i b i l i t y of f r ac tu re o r f a i l u r e
of t he composite under impact, the complete s t r e s s matrix a t a point
must be known, as well as the f a i l u r e c r i t e r i a f o r t h e mater ia l .
- 80 -
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CAPTIONS FOR F I G U R E S
3
4
5
6
Geometry of a two dimensional wave i n a mult i -ply p l a t e
Velocity sur faces versus wave d i r ec t ion f o r var ious
p l y lay-up angles; 55% graphi te fiber/epoxy ma t r ix
(Moon, 1972)
Direct ion of p a r t i c l e motion versus wave normal f o r
var ious p l y lay-up angles; 55% graphi te fiber/epoxy
matr ix (Moon, 19'72)
Wave sur faces f o r mult i -ply p l a t e s ; a)
angle, b ) 215' f i b e r lay-up angle (Moon, 1972)
0' f i b e r lay-up
0 Wave sur faces f o r mult i -ply p l a t e s ; a) +30 f i b e r lay-up
angle , b) +4S0 lay-up angle (Moon, 1972)
Flexural wave d ispers ion r e l a t i o n s i n an an iso t ropic p l a t e
(Mindlin's theory); 55% graphi te fiber/epoxy matrix mult i -
p l y p l a t e , +45 f i b e r lay-up angle. 0
Surface waves and edge waves i n s o l i d s
Dis tor t ion o f an i n i t i a l l y shaped t rapezoida l pulse
due t o wave d ispers ion e .g . longi tudina l waves pro-
pagating across o r down t h e f i b e r s of a un id i r ec t iona l
f i b e r composite material
- 97 -
9
10
11
1 2
13
14
Distor t ion of an i n i t i a l l y shaped pulse due t o wave
dispers ion e .g , shear wave propagating down the f i b e r s
of a unid i rec t iona l f i b e r composite material
Approximate dispers ion r e l a t ions f o r longi tudinal waves
i n boron fiber/aluminum matrix rods f o r various or ien ta t ions
of t h e f i b e r s t o the rod ax is (Pot t inger , 1970), mater ia l
dispers ion not included .
Sketch o f dispers ion r e l a t i o n f o r longi tudinal o r shear waves
propagating normal t o the layers of a composite of a l t e r -
nat ing i so t rop ic layers , E q . (26) .
Comparison of exact dispers ion r e l a t i o n s ( so l id l i nes ) w i t h
t he microcontinuum theory of Sun e t a l . (1968) f o r various
shear modulus ratios; a) shear waves propagating i n t h e
d i rec t ion of t he layer ing, b) longi tudinal waves propa- I
gat ing i n t h e d i r ec t ion of the layer ing
Shock wave speed versus p a r t i c l e ve loc i ty (Hugoniot curve)
f o r a mixture of AR 0
(Munson and Schuler, 1970)
p a r t i c l e s i n an epoxy matrix 2 3
Experimental speeds of longi tudinal waves i n steel fiber/epoxy
matrix rods fo r various volume f r ac t ions of s teel , (Nevi11
e t a l . , 1972) ; waves t r a v e l l i n g along t h e f i b e r s
- 98 -
15
16
17
18
19
The e f f e c t of f ab r i ca t ion on shock wave spa11 damage
i n a boron fiber/aluminum matrix composite (Schuster
and Reed, 1969); a) brazed composite, b ) d i f fus ion
bonded composite
Experimental dispers ion r e l a t i o n f o r longi tudinal waves
propagating down t h e f i b e r s ( h a y e t a l . , 1968); a) graphi te
f i b e r (Thornel) re inforced carbon phenolic composite,
b) boron f i b e r re inforced carbon phenolic composite.
Experimental dispers ion r e l a t ions f o r waves i n boron f i b e r /
epoxy matrix composite (Tauchert and Guzelsu, 1972)
upper f igu re - longi tudinal waves normal t o the f i b e r s ;
lower f igu re - shear waves, x ax is is along t h e f i b e r s 3
Experimental dispers ion r e l a t i o n f o r longi tudinal waves
i n a tungsten fiber/aluminum matrix composite (Sutherland
and Lingle, 1972) lower curve shows second branch and a
cutoff frequency around 4 MHZ
Ortho t ropic phot oe 1 as t i c i t y experiment showing an anis0 t ropi c
extensional wave i n a p l a t e loaded with a lead azide charge
i n t h e center (Dally, e t a l . , 1971), compare with Figures
4, 27.
- 99 -
20
21
22
23
24
25
26
Dynamic s t r e s s - s t r a i n curves f o r steel fiber/epoxy matrix
composite under various s t r a i n rates (Sierakowski e t a l . ,
1970a) tests were conducted using compressional waves along
the f i b e r s
Impact damage i n a graphi te fiber/epoxy matrix p l a t e
( . 2 5 e m , 0 . 1 inches th i ck ) showing back face s p l i t t i n g
f o r 0 . 6 4 cm (1/4 inches) diameter steel b a l l s a t
115 m/sec i n i t i a l ve loc i ty .
+45 (Novak and Preston, 1972)
Ply lay-up angles t45', O o ,
0
Wave sur faces generated by a l i n e impact on a an iso t ropic
h a l f space (Kraut, 1963)
Contact times based on Hertzian model ca lcu la t ions f o r t he
impact of i c e b a l l s and g ran i t e spheres on graphi te f i b e r /
epoxy matrix h a l f space
S t a t i c experimental contact force r e l a t i o n f o r a 3/8 inch
diameter steel b a l l on graphi te fiber/epoxy matrix composite,
normal t o t h e f i b e r d i r ec t ion
S ta t i c experimental contact force r e l a t i o n f o r a 1/4 inch
diameter steel b a l l on boron fiber/aluminum matrix composite,
normal t o t h e f i b e r d i r ec t ion
Geometry of impact with a p l a t e , showing t h e effect of
motion o f t h e s t r u c t u r e
- 100 -
27
28
29
S t r e s s contours f o r t he membrane stress 1 / 2 ( t + t )
a f t e r impact f o r a 55% graphi te fiber/epoxy matrix
p l a t e . Comparison of a) O o , and b) 245 p ly lay-up
angle cases (Moon, 1972)
1 1 33
0
Stress contours f o r t he lowest f l exura l wave stress
1 /2 ( t
epoxy matrix composite p l a t e . Comparison o f a) t15
and b) t 4 5 p ly lay-up angle cases (Moon, 1972)
+ t 3 3 ) af ter impact f o r a 55% graphi te f i b e r / 1 1
0
0
Three dimensional p l o t of t h e lowest f l exura l wave,
1 /2 ( t + t ) , and a q u a r t e r plane o f t he p l a t e f o r
a 55% graphi te fiber/epoxy matrix composite p l a t e with
+45O p ly lay-up angles ( f i b e r s along diagonals)